Studying Turbulence Using Numerical Simulation

Studying Numerical

Proceedings

Center

Turbulence Simulation

of the

for

1987

Summer

Turbulence

Report December

Using Databases

Research

CTR-S87 1987

G3/3q

N/ A Ames

Research

Program

lee-| 3121 lac, las 01_17"T7

.

N_S Center

Stanford

University

Studying Numerical

Proceedings

Center

Turbulence Simulation

of the

for

1987

Using Databases

Summer

Turbulence

Program

Research

Report CTR-S87 December 1987

N/ A Ames

Research

Center

Stanford

University

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CONTENTS Preface

Stochastic

Decomposition/Chaos/Bifurcation

Group.

Overview Stochastic

estimation

R. J. ADRIAN,

of conditional

P. MOIN

eddies

in turbulent

channel

flow.

and R. D. MOSER.

7

A low dimensional dynamical system forthe wall layer N. AUBRY L. R. KEEFE. Self

similarity

flows.

of two

J. C. R. HUNT,

Ejection

mechanisms

J. JIMENEZ, K-space

point

P.

correlations

P. MOIN,

in the MOIN,

R. D. MOSER

sublayer

R. D.

in wall

bounded and

of a turbulent

MOSER

and

and 21

turbulent

P. R. SPALART. channel.

L. R.

KEEFE.

37

Group

Overview

49

EDQNM

closure:

"A

homogeneous

simulation

quasi-homogeneous simulation to disprove K. SQUIRES, and J. H. FERZIGER. Validation of EDQNM P. ORLANDI. Angular J. H.

distribution FERZIGER,

Big Whorls Carry A. A. WRAY. Study

for subgrid

it."

support

J.P.

it,"

"A

BERTOGLIO,

53

supergrid

of turbulence in wave J. P. BERTOGLIO. little

whorls.

models.

J. C.

R.

space.

COLEMAN,

G.

71 J. C.

HUNT,

BUELL,

and 77

spectral

homogeneous modeling.

Modeling scalar A. YOSHIZAWA.

and

to

63

of one-dimensional

stresses in split-spectrum

Scalar

25

flux

and

dynamic

equations

anisotropic turbulence: R. SCHIESTEL. the

energy

and

of the

Reynolds

Application

to 95

dissipation

equations. 109

Transport/Reacting

Flows

Group

Overview

113

Direct numerical simulation of buoyantly driven WM. T. ASHURST and M. M. ROGERS. Premixed LAND,

turbulent J.

H.

flame

FERZIGER

calculation. and M. M.

turbulence.

S. EL-TAHRY, ROGERS. iii

117 C.

J.

RUT121

Modeling of turbulent transport NINGS and T. MOREL. Analysis and J.

of non-premixed C. HILL.

A statistical ticity W.

on

Reynolds

Stress

M.J.

JEN-

reacting

flows.

A.D.

LEONARD 141

of the

direct

KOLLMANN

process.

133

turbulent

investigation

based

as a volume

single-point

numerical

and

K.

pdf

of velocity

simulations.

M.

and

vor-

MORTAZAVI,

147

SQUmES.

Modeling

Group

Overview On

155

local

approximations

models.

of the

P. BRADWHAW,

N.

pressure-strain N.

MANSOUR

term and

in turbulence

U.

PIOMELLI.

159

Local structure of intercomponent energy transfer in homogeneous turbulent shear flow. J. G. BRAUSSER and M. J. LEE. A general J.

C.

R.

form for the dissipation length HUNT, P. S. SPALART and

Test code models.

for the M. W.

assessment RUBESIN,

scale in turbulent N. N. MANSOUR.

and improvement J. R. VIEGAS, D.

shear

165

flows. 179

of Reynolds VANDROMME

stress and H.

HA MINrl. Reynolds N. The C.

N. decay G.

185

stress

models

MANSOUR, of isotropic SPEZIALE,

N.

Evaluation of a theory K. SHARIFF. Relations between M. WOLFSHTEIN Turbulence

of homogeneous

and

J.

Y.

turbulence.

SHIH, 191

turbulence in a rapidly N. MANSOUR, and R. for pressure-strain

rotating frame. S. ROGALLO.

rate.

J.

WEINSTOCK,

205 and 213

two-point correlations and S. K. LELE.

Structure

T.-H.

CHEN.

and

pressure

strain

terms. 221

Group

Overview

231

Shear-layer structures P. H. ALFREDSSON The K.

simulation BREUER,

of coherent M.

T.

Conditionally-averaged Y.

G.

Structure and J.

in near-wall and J. KIM.

structures U.

of turbulent KIM.

shear

in a laminar

and

P.

V.

JOHANSSON,

and

boundary

layer. 253

S. SPALART.

in wall-bounded

PIOMELLI flows.

A.

237

structures

LANDAHL

GUEZENNEC,

turbulence.

turbulent

flows. 263

J. KIM.

A. K. M.

F. HUSSAIN,

J.

JEONG 273

iv

Inflectional

instabilities

flows. P.

S.

J.D.

in the

wall region

SWEARINGEN,

R.

Wave-growth S.

F.

turbulent

shear and

BLACKWELDER

291

SPALART.

Active layer model for wall-bounded J. KIM and P. S. SPALART.

D.

of bounded

associated

with M.

HENNINGSON,

turbulence.

T.

LANDAHL,

297

turbulent

T.

M.

spot

LANDAHL

in plane

and

J.

Poiseuille

flow.

KIM.

303

Appendix Coherent

structure.

P.

Coherent

structures

- comments

Coherent

structures

and

Coherent structures S. J. KLINZ. Coherent

structures.

BRADSHAW.

313 on mechanisms.

dynamical

and modeling:

systems. some

J.

J.

C. R.

HUNT.

JIMENEZ.

background

315 323

comments. 325

C. G.

SPEZIALE.

329

v

Center

for

Turbulence

Proceedings

of the

Research

Summer

Program

1987

Preface The first the

four

Summer

week

Program

period

July

of the

Center

13 to August

for Turbulence

7, 1987.

databases obtained from direct numerical of turbulence physics and modeling. Thirty-three research

participants

proposals.

* Stochastic

from

They

were

eight

The

divided

program

simulations countries into

Research

was held

focused

of turbulent

were

selected

during

on the flows,

on the

use of

for study

basis

of their

five groups:

decomposition/chaos/bifurcation

• Two-point closure (or k-space) modeling • Scalar transport/reacting flows • Reynolds

stress

modeling

• Structure

of turbulent

In addition to these ticipation of Robert seminars

participants, Kraichnan

presented

by leading

The databases neous turbulence bulent

consisted subjected

channel

considered

boundary

flow

were

and

low

and

layers. the program benefited from the week long parEvgeny Novikov. There were 8 tutorials and 4

experts

in different

areas

of decaying and forced to strain, homogeneous the

turbulent

to moderate;

research.

isotropic turbulence, homogeshear flow, fully developed tur-

boundary

Taylor

of turbulence

layer.

micro-scale

The

Reynolds

Reynolds

number

50 in homogeneous flows and Reynolds number based on free-stream boundary layer thickness of about 3000-10000. Most simulation fields of passive The in the ideas be

scalar

reports

that

resulted

following

pages.

This

is an account

and

models

in due

together,

Fluid

Twenty abstracts presented at this In our opinion, of conceptual Summer

It

of many Dynamics

ideas Program

1987

Therefore,

the

reports

written

of these

Summer

velocity included

between

participants

demonstrated

a powerful tool in turbulence catalysts for interaction among

to the

the

but cases,

and fields

occurred

the

with

varied

be

the

research, and the databases turbulence researchers.

should by the clustered

group.

American Nov

setting

backgrounds

of using

are

simulation proved

Parviz

of

continued

journals group

summer

to be a valuable

viability

results

Oregon,

during

study

will

of that at

included

the

appropriate research

coordinator

are

intensive,

studies

in Eugene

accomplished proved

the

of each

projects

program

term,

in most that

by the

Meeting

Program

summer

of a short

is expected

Division

based on the work meeting. the

the

will be submitted

volume

by an overview

reporting

from

flows.

results

In this

preceded

Society

The

the

authors.

Timely cal

of turbulent

as preliminary. course,

individual

of about

as well as velocity.

research

considered

and,

contaminant

numbers

Physi-

22-24,

1987.

program

were

for exchange and

interests.

databases to be

as

effective

Moin

William C. Reynolds John Kim

Center for Proceedings

Turbulence Research of the Summer Program

3 1987

Overview of research by decomposition/chaos/bifurcation This

group

consisted

of four loosely

of understanding

the

the

channel

un-processed

two-point The

correlation

invited

Ronald

Dr.

Nadine

Dr.

Julian

local

Dr.

(University

(Cornell

University)

Professor

(Stanford

from

the

velocity

wall

layer.

coworkers

had

was

A similar already

direction

found.

These

applications The

for velocity form

shown

Aubry,

results where

systems Holmes,

perimental

f(y/yl).

Using

layer. obtained

used ited

they

methods

be

two-point

very

in shear

of Herzog these

a highly

of dynamical

intermittency

(1986)

which

system was

A similar analysis was performed from simulation databases (Moin those

of Aubry

intermittency, the

character

et al. rather of the

the

PRECEDING

PAGE

Hunt

layers.

and Com-

(e.g,

of the

to analyze with

the

were used

appeared

the

in other

events

work the

ex-

of a turbulent

physically

Stokes

results.

in Aubry's

from

region

Navier

bursting

was

previous

obtained wall

which

solution

and

In their

near

shear

see below).

tensor

turbulence.

in the

on

as roll-

equations

The

results

in the

and exhib-

sublayer.

at the CTR using the eigenfunctions computed & Moser 1987). The results were different from limit

fixed

intermittency

log layer.

modeling

eigenfunctions

eigenfunctions,

In particular, than

are used

layer

theory

of the

boundary

in turbulence

used

associated

corre-

portion

dependence

correlation

(1986)

truncated

free

point

generated

a large

a strong

data

of wall

& Stone

Employing

useful

two-point

representation

numerically in the

the two-point v-_ at the

shear reduces the correlation length of The variation of eddy scales in the

and

correlation

of the spatial

that

intensity

throughout for R,,,

f is linear

investigated,

should

measurements

cells,

Spain)

states

by the

was obtained

that

also

Lumley

boundary

& UAM-IBM,

Ames)

correlations

normalized

to be valid

collapse

shown

was

eigenfunctions

dynamical

velocity

1987).

Madrid,

& NASA

parison with their results clearly shows that the normal velocity, in the normal direction. spanwise

or the

& Moser

Research)

University

v, when

is of the

hypothesis

boundary

1987)

used

Ames)

hypothesis

of normal this

Politecnica,

for Turbulence

(NASA

self-similarity

lations,

Moser

it (Moin

All projects

of Cambridge)

(Center

Moin

D. Moser

furthest

flows.

objective

were:

Parviz

correlation

Moin,

the common

of Illinois)

(Universidad

R. Keefe

Robert

Hunt's

(University

Jimenez

participants

Laurence

Dr.

J. Adrian

Aubry

Javier

turbulent

(Kim, from

with

were:

C. R. Hunt

Professor

database

computed

stochastic group

projects

of wall-bounded

flow

tensor

participants

Professor

The

mechanics

inter-related

the

cycle

point

is different

BLANK

behavior

behavior and

NOT

was

found

observed previously.

it is significantly

FILMED

just

prior

to

As a result more

sensitive

4 to the bifurcation

or eddy

viscosity

work

in relating

dynamical

work

is required

to determine

parameters

and

Two-point work

of Moin,

dies.

The

other

parameter.

systems the

In view

of the

theory

to the

structure

sensitivity

of the

results

significance

of this

of turbulence, to various

further

computational

inputs.

correlation

data

Adrian

& Kim

previous

work

was

also

used

(1987)

applied

on

only

to extend,

stochastic

in planes

to

three

estimation

transverse

dimensions,

the

of conditional

to channel

ed-

flow.

With

stochastic correlation

estimation, one approximates conditional averages using the two-point tensor. In addition, the theory was extended to include specification of

conditions

at more

verification tation

that

of conditional

of the

structures.

that

occur

layers

eddies This

in the

in the

one point.

estimation

It was

shown

field

(see

investigation

represen-

stochastic

estimates

representations Using

of the

instantaneous

the

asymmetric

conditions

obtained

a simplified

vortical

model

structures

was the

an accurate

two-point

of generating

below),

of inclined

of this

provides

that

reasonable flow field.

consisted

result

indeed

is capable

instantaneous

which

also

provide

technique

instantaneous

proposed

An important

stochastic eddies.

conditional

flow

was

than

linear

structures

of the

from

shear

shear

layers

surrounding

each

shear

layer. Perhaps

the

turbulent

most

channel

layers.

The

dramatic

flow contains

shear

layers

importantly,

spanwise

vorticity

resembled

those

in channel to

followed

in time

observations sublayer to the

normal (1987)

shear same

interactions

with

an interesting side

are

and

the

discovery

that

visible,

shear

highly

vorticity

was

size

shear

structures).

of the channel,

solution and

average wall shear stress of the of the 2-D non-linear solutions

three

proposed

of the

numerical

wall,

protruding

depicted

and

from

in the

shear

the

This

One which

essentially turbulent and the

box,

layers

Based

attempts

were

absence

two-dimensional

of this flow

latter

waves. made study

turbulence

on the

values

the

of complex

three-dimensional

layer falls between the 3-D turbulent solutions.

from

equivalent

Schlichting

the

were

on these

ejection

by products

displayed

strongly mechanism

is essentially

futile (in

of

Although

shear

vorticity

in isolation of the

The directly.

of 2-D Tollmien

computational layer

direction,

generation

model

plots

solutions.

case.

to explain

instability

flow

the

was observed

layers.

in contour

numerical

dimensional,

process

for the

of one

layers,

two-dimensional

of the

the other

was

of strong,

spanwise

shear

to the

generation

model

dynamics

group

two-dimensional

as in the

this

responsible

by reducing the

layers

the production

mechanism

to study

on one

in planes

and

density

of strong

of these

in Jimenez's

be the

in this

layers. Apparently the dominance of these shear layers, at numbers considered here, has been overlooked previously. patterns

a simple and

Finally,

was

the

flow these

appears

a high

are regions

wall region into the outer least for the low Reynolds More

observation

other.

The

characteristic

Parviz

Moin

Overview

of the stochastic

decomposition/chaos

group

5

REFERENCES AUBRY,

N.,

HOLMES,

of coherent ley School Cornell

Phys. KIM,

3., channel

MOIN,

P.

Ithaca,

1987

Fluids. MOIN,

30,

and

Bifurcation

AND

E.

STONE,

of a turbulent

Aerospace

Engineering

1986

The

boundary . Report

dynamics layer.

No.

Sib-

FDA-86-15.

and

bursting

in two-dimensional

Poiseuille

flow

3644.

P. g: MOSER,

& MOSER,

L.,

wall region

NY.

R.

flow at low Reynolds

in a channel.

J.

LUMLEY,

in the

of Mechanical

Univ. J.

JIMENEZ,

P.,

structures

R. D.

Submitted

1987

D. 1987 number.

Turbulence J. Fluid

Characteristic

for publication.

statistics Mech.

eddy

177,

in fully

developed

133-166.

decomposition

of turbulence

.

Center

for

Turbulence

Proceedings

of the

N88-23087

Research Summer

Program

Stochastic eddies By

1.

R.

J.

estimation in turbulent

ADRIAN

of conditional channel flow

1, P. MOIN

2,3 AND

R.

D.

MOSER

2

Background In two recent

simulation (in

the

studies,

data

where

stochastic

bases

(Adrian

mean-square

points

sense)

certain

data

estimation

& Moin estimate

are given

square

error.

data,

E(x).

used,

and

velocity

E(x)

and

is the

latter

In both

study,

studies

studies

inhomogeneous Moser

estimated

probability

(1987)

from

contain

two

and

tensor,

the stochastic

any

of points.

and

was obtained

estimation

Consider any array of points conditional eddies defined

more

only

et al.

the

used

University NASA-Ames

3

Stanford

was

of the a

vector.

the

new

elements: using

yW,rz,vz)

data

threetwo-point

and

was

fornmlation

(x j, x2,..., by

events.

recently

was

base

dimensional

data,

E.

The

computed

utilized

was extended

x,_ ). The

structure

conditional

to include averages

U(XN))

__

self-similar

f is approximately

the

(y/yl)

of other eddies close to or far from the wall. one of these mechanisms is dominant, while to be considered

layer

(yl

v

2-

case,

There shear

and

structure ,ou, at y, (_),

in that

at the

effect.

Introduction The

But

two-point Yl

that

to half

has

boundary

3

-/(7, )

= 1, and

similarity

or 'splatting'

y,

to have

of self

an important

for

points

channel

7000

SPALART

(1986)

turbulent

from

layers.

at two

are

< 1, f(O)

kind

Spalart

and

f _ 2(y/yl) The

and

P. R.

bounding

in terms

for yl ranging

3 AND

of turbulence

,,(v)v(u,)

where

MOSER

the

wall

component

the in of

26

J. C. R.

Hunt,

P. Moin,

R.

D. Moser

and

P. R.

Spalart • ='"y/ylV L VL + VL (COMBINED EFFECT

W_ITH_ j/IMAGE)vL

/ i / "

_l

/_,""

\

I \

l I I

\ \ I

\ I

% ",

I

IL '

ture

1.

1

(b)

Diagram

of an updraft

the

velocity

//1.

The

are

L' and

eddies,

blocking research

convection

a density form

upper

point),

The

sensitive

effect,

of the

on shear

the

be a large at y. Then

of the

combined

surface

(where

has

v(y)v(yl)

_ f(_/)

The

(L) centered

the velocity

is valid

when

explanation with maximum

at Yl, v(yl

at

height

velocity

the

"images"

large

in inviscid

and

small

a free

amplifies

surface (i.e.,

less than

with

the

in thermal

or turbulence leads

to a self-

normalized

at the

of the

(1.1)

the turbulence

aid of figure

at Yz and

v)

flow).

as occur

y < Yz

9z is much

effect

(such

v2(yl)

yl

velocity

(which

bl___ocking effect

_, y

) ._ VL. The

(Their

of the

of shear

layers,

with

is given

y.

of both.

near

how

R,,,_ -

SFBL

between at height

v = 0, even

as a ratio

for the

(b) the relation

boundary

shown

struc-

VL, is centered

effects

expressed

(a) typical

velocity

vertical

relative

ground,

updraft,

effect

or in turbulence

layer)

the

is centered

when

boundary.

eddy

L, with

vs

to the

near

"image"

free turbulent

boundaries

inversion

the

velocity of the

motion

eddy

are shown

and

for R,_

theory

far from

large

most

between

similar

with

eddy plus

The

profiles

images

is the

the

y. S,

S') The

Recent near

or "thermal"

eddy

of their

turbulence and

to illustrate

at Yl and

small

--

IMAGE EDDIES

(a) FIGURE

/

a small

small

eddy

scale

1. Let there eddy

centered

at y is small

Self

Similarity

of Two Point

Correlations

r-I

RUN 2

27 -zi/L 350

V

3

30



5

25

>

8

502




0

g er

(a) _

I

,

,

f

1

2

3

1.0

.5 V

o

0

5t

-1.0

(b)

0

r3/a 3 FIGURE in z (r3) of Yl.

6.

Normalized

cross

turbulent

plane

from (a) v(y,

correlations channel

ra),v(yz,O)/v(y,O)v(yl,0),

value of a3 is chosen through 1/e.

for each

curve

of velocities

flow (Moin (b) v(y,

to be the

at y and Yz with

& Moser,

1987)

separation

for various

ra),u(yl,0)/v(y,O)u(yz,O).

r3 location

at which

the curve

values The passes

34

J.

C. R.

Hunt,

P. Moin,

R.

D. Moser

and

P.

R. Spalart

.6 .5 .4 _O

"_.3

J

0.71 0.49

.2 0.11 N

.1

0.23

(a)

0

0.03

0.11

1

0.23

.1

L

" (b)

0.03 i

0

.2

.4

.6

.8

1.0

Y/Y1 FIGURE (figure

The

7. 6a),

of the

to scale

are

as functions

as the variation

plotted value

form

scale

of the

emphasizes

for values

this

of Yl/6

10. This

log layer.

show

firstly,

negligible

is of the

+ 7)

Ya, where

results

centerline,

A satisfactory

in the

y and

layers.

OU +

0.3y +

as used

approximate

8)

a + ,_ (1.4 0--_

(a)

and

a3 increases this

when

in the introduction.

Yl,

different

structure

true

how

for

spanwise quite

It is not show

=

and

Rv,

6b).

of r3/a3

of the

y+.

of Y/Yl

Rv,_ (figure

of r3 where

of the

spanwise

correlation

of a3 as a function

(b)) as used

curves

defined small

Values

curve

this

yl.

the

variation particular

invariance Ya/g

as is

is found

_ 1.0. In Figure

order would

7, we

of 9 wall units be consistent

fit could

be (figure

Self Similarity

of Two Point

35

Correlations

.3

so+_ 20

.1

10 0 -

0

I

i

I

I

.1

.2

.3

.4

I

.5

y/6 I

I

I

I

I

I

I

I

I

0

10

20

30

40

50

60

70

80

I

90

y+

FIGURE

8.

Values

the reference

of a3 as used

91- × - simulation

height

This

value

scale

L_ (based

to scale

of a + is of the

same

on v 2 in the

(figures

results,

order

but

log layer) L,

R_

a little

6a and

o - the

model

9rearer

than

7a),

as a function (1

4 du+

aa = 7 + _-'-a_-_/

the dissipation

of _-I

•

length

where

-

>- O.18y ._-23 / 2

Other

results

channeling

of the

as great 3.

have

been spanwise

and

The preliminary the

of these

spacing,

The

we have aspect scale must results

a. Comparison b.

The still

though

They

the

value

also

show

equally

strong

of aa for R,_, is about

twice

effects expect

work

show

functions

that

the

of correlation

this

by small structure

further

that

correlation

similar

shows

form

flows;

structure;

results

two-point

self

results the

for Ru,,,R,_.

as for R,_.

Implication

layer,

computed

that

structures take both

with

c. Further theoretical shear near a wall.

and

for spacing the

structure very these

eddy

inhomogeneous have

by using

structure

is largely

study

laboratory

and

atmospheric

and pressure gradient self similar two point

calculations

should

are

to tile

The

nature

theory.

equally wall.

is quite controlled

boundary

forms.

rapid-distortion

of blocking normal

turbulent

self sinfilar

close to the wall. So any effects into account.

further

of curvature to see these

may

be inferred

of shear

functions

of eddy

suggest

can

effects

found

even in the

function

But

different shear

theory

for

in

for spanwise

in these

by the

The

important

and

shear

perhaps

the

turbulent

measurement. should be investigated. correlations.

be done

using

RDT

Theory

We would for

uniform

30 d.

J. The

two

nents

point

and

C. R. Hunt,

P. Moin,

correlation

in other

functions

direction

R.

D. Moser

are

(eg.

likely

for the

and

P. R.

Spalart

to be self similar

spanwise,

for other

compo-

z, spacing).

REFERENCES HUNT

J.

C. R.

HUNT

J.

C. R.,

1984

J. Fhtid

KAIMAL

Mech.

J.

C.

138,

161-184.

g5 GAYNOR

J.E.

1987

Quart

J. Roy

Met.

Jour.

(accepted). KIM,

J.,

MOIN,

channel LEE,

M.,

KIM,

7-9 1987, MOIN,

P.

1410. TOWNSEND

MOSER,

J.

& MOIN,

Toulouse,

&_ MOSER,

in a channel. SPALART,

P. _

R.

flow at low Reynolds

P.R. NASA A.A.

D.

1987

number.

P.

1987

Turbulence J. Fluid

6th Turbulent

statistics Mech. Shear

177,

in fully

developed

133-166.

Flows

Symposium,

Sept

France. R.

D.

1987

Characteristic

eddy

decomposition

of turbulence

Ill preparation. 1986

Direct

simulation

TM-89407. 1976

Stvuctune

(J.

Fluid

of a turbulent Mech.

of Turbulent

boundary

layer

up to Re =

ill press) Shear

Flows.

Cambridge

Univ.

Press.

N88-23090 Center

for

Turbulence

Proceedings

By

of the

J.

It

JIMENEZ

1, P.

long

been

layers

accepted

picture The

for

the

velocity

are indications

structure

that

the

ejections layer.

processes

originate,

Jimenez

(1987),

channel

flow,

found

a simple

ejections

of vorticity,

and

of the

associated

wall

into

ticity

the

into into

the

viscosity

(figure

in a 2-D

flow

usually

flow.

thin

1).

core

is _.,

and

that

complete

this

mechanisms

contributions picture

would

survive

in fully

A particularly tribute

to

shown flow

appealing

the

origin

in Jimenez is tile

of the

(1987)

viscous

ejections

that

sublayer,

general

behavior

was

it can

Tollmien-Schlichting wave

train

waves,

is characterized

separated

by regions

and

vorticity

weak

1 Universidad 2 Stanford

Politecnica,

by

of weaker, generates Madrid,

University

3 NASA

Ames

4

for

Center

(T-S)

Turbulence

2-D

Research

the

solution of the

channels,

basic

ejection

reasonably

is that kind

a succession

described

dissipated

by

include that

could

con-

since

instability

for to be

train peaks alternation

local

updrafts,

corresponding

to stagnation

and

UAM-IBM

Scientific

Spain.

flows, of it

especially

occurring

this

aspect

mechanism

This

of the

channel some

vorticity

Center,

any

which

in (Herbert,

vorticity.

present

expected

of a periodic

of strong

or even negative

vortices,

it is vor-

from

be expected

phenomena

the

of their

component

different

same

from

is ejected, part

not

outer

of a two.

away

vorticity

it was

viscous

to spontaneous

fluid,

vorticity

and

in the

behavior

three-dimensional

in natural

where

Cantwell,

of the

rise

to check whether situations.

that

site of the

e.g.,

active

release

by hairpin

developed

layer

it is eventually

As such,

the

of the

the

tile only

imately independent of tile three-dimensional part of natural boundary layers. The

once

A generally

wall

behavior

momentum

in

not understood,

those the

is essentially

_%.

(see

giving

where

that

workshop was fully turbulent

possibility

the

periodically

induction

w_, and

but one of our goals during this could still be useful in describing

case,

process

involving

from

low

channel,

be stressed

1967).

is, however, from

4

turbulence

the

production.

numerically

which

of the

It should

from

mechanism

2-D

layers

et al.,

different

KEEFE

bounded

(Kline

controlling

studying

In that

shear

laminar

accepted

important

core

long

while

L. R.

ejected

ejection

are

Recently,

and

are

the initial

3 AND

of wall

of turbulence

triggers

dimensional

stretched

MOSER

streaks

fraction

that

sublayer, where these parts of the boundary

D.

the sublayer channel

is three-dimensional

low

a large

in

2,3, It.

that

channels

is that

mechanism

there

MOIN

recognized

and

responsible

and

1987

mechanisms of a turbulent

has

1981).

Prograrn

Ejection

boundary are

37

Research

Summer

it was the

2-D

approx-

in the

outer

of nonlinear 1976).

This

at

wall,

the

of strong points

in

38

Jimenez,

Moin,

Moser

and

Keefe

L jf

111

FIGURE The

1.

Two

periodic

equivalent

a frame from

ejections to a limit

to right, frames

and

time,

from

from

the

of reference

the

into

from cycle

spans

wall,

this situation a wave

dimensional

a limit

top

moving

producing

number cycle,

the of the

lower

depends

the

a = 1.0,

train

the rise

in the

to bottom. wall

to the

Our that, the

first some

channel

codes

are

step initial

channel

was

conditions

flow described spectral,

they

shear

uniform

wave periodic

accuracy

from

Jimenez

essentially

which

Flow

primary

layers

tend

(see

Above

waves

are

runs from

left

T-S

train

to draw

figure

becomes

ejections

2).

Re = Uh/u unstable

described

Throughout

the

Moin

with

channel.

Schlichting

1987).

wave.

Each

vorticity

away

this

of the

original

(1987)

were

& Moser,

1987).

independent,

and

stability

= 5500, and

of

and

for

bifurcates At

a higher

behavior (a torus). with the same mass

paper 2-D

used

The

above.

complicated dynamical of a parabolic profile

width.

of a 2-D

centerline.

train,

number.

in (Kim, are

move

Reynolds

half

to check

Axes channel

Re = 9100, it bifurcates again into more Here, U refers to the center-line velocity flux, and h, to the non-dimensionalization.

Jimenez,

protruding

layer

Tolhnien-

(from

the wave

to the

boundary

of nonlinear

system

with

on the

giving

ejection

we will

calculations.

with

a 2-D

Although

both

differ

in many

use

this

To do version numerical important

of

Ejection

FIGURE

2.

The

basic

mechanisms

mechanism

for the

the creation of an updraft between unequal magnitude) in the sublayer.

details, less,

including

the

cycle

(Re

nature

different

results

of both

= 7000),

of the

and

results

the wall stress, and plotting accuracy. The

next

step

lation, and

Re

velocity

thai. reference experimental The

first

common manner (figure

which 3).

described evolution There

observation

of this is strongly

To our the

with

flow,

reminiscent

knowledge,

formation

is that

channel this

of thin

those in the 2-D with an average

of the

of 1 to 3h.

order

only

exist,

between

values

resolved

nonlinear

mass

T-S

For that, simulation

numerical in the

number flux.

of

to within

channel flows. the numerical

Reynolds same

qualitative

agreed

as 47rh periodic the

of the

as equilibrium

6.5h. Also, the channel to level off at a distance

of z-vorticity

they

features

simu-

z direction,

is based It was

on the shown

are indeed

protrude

observed

observation,

of high

from in the

although

vorticity

the

in

a very

wall

To begin separation

Kim

magnitude

while

for wavelengths

with, the between the

2-D

(1987)

as part

"wavelength" consecutive nonlinear

in a range

in a

2-D calculations had of the

in the neighborhood of a channel between the structures observed

wall units),

solutions,

layers

that

is a new

layers

calculations. longitudinal

(200-600

thin

and

of an isolated "hairpin" vortex are some important differences

channel and to be shorter, can

profile

the

a limit

quantitative

instability,

(or

Neverthe-

included

not only

the

is a fully

The

sign

integration.

of similarity

is defined

of opposite

is

its statistical properties are in good agreement with those of and we will consider it here as a "natural" turbulent channel.

surprising

feature

of the

This

z direction.

of a parabolic

that flows,

but

present in natural fields extracted from 1987).

of vorticity

comparisons cases,

similar,

as a result

which

in tile

in the

The

In both

were

ejection

vortices

used

the degree

& Moser,

of a channel

periodic

of spanwise

= 9200).

codes

of its oscillations

Moin

= 4200,

centerline

(Re

both

39

dimensional

variables

layers of z-vorticity time series of flow

in (Kim,

as 4rrh/3

a torus

from

two

sublayer

did check in detail.

was to investigate

waves and thin we used a short described

a pair

dependent

codes

in the

wall. in the seems features

T-S

between

waves 4b

and

layers penetrate less into the core of the channel, appearing of 0.3h. (35 wall units) away from the wall, while the 2-D

40

Jirnenez,

Moin,

Mosey

and

Keefe

Y

FIGURE

3.

Lateral

natural,

fully

developed

structures

in the

_o_ = 1,2;

solid:

view

2-D

of a train

three

of projecting

dimensional

calculations.

shear

channel

Dotted

Note

of z-vorticity the

to _o, = -1,0;

at the

wall

is w_ = 7.67.

_____.._._ _..__..; ::: .............. =_.z .... _-... ---

__..-;_,.

vorticity

in a

similarity

correspond

wz = 3 to 17. Average

lines

layers

flow.

to the dashed:

Z

,-_= F 4.

FIGURE

(y+ "A"

Distribution

= 6).

These

solutions some core. The three

extend

main

at the wall

least

higher

characteristic can

be used

(figure

4).

velocity, at the

than

, and and

They

appear

convection

mt_ss flux. velocity

an average with

U refers

This last

of the

the

2-D

ejection

angle

a convection

number nonlinear

centerline

the

into

velocity

natural than

channel

They

Although

good the

are

appear

channel their

is at

agreement significance

a

spots motion

center

laminar

to

with

These

to follow

is the

or about

vorticity

the

of the

there into

0.2h,

of 60% of the

velocity

marked

hand, deeper

spanwise

and

sublayer

line

of a few degrees.

layers

is in surprisingly waves.

other

wall).

to extend

protruding

to the

at the

in which and

in the

of no more

as much

value,

3. The

do extend

layers

viscous

3.

On the

that

extent

spots

wall

count

to move

where

twice

in figure

line.

shear

in the

in figure

layers

the

"hot"

its average

S-shape

or 0.47U, same

in elliptical

to detect

center

a spanwise

about

vorticity

section

of weaker

ix

I

structures

cross

is that

with

at y+ = 6 (and

be rooted 25%

however,

structures,

spanwise

channel

flow,

_-:

I

of the

of the

way to the natural

difference,

units,

of high

roots

position

all the in the

dimensional

35 wall

are the

to the

evidence,

I"

of spots

spots

corresponds

.._-.-y.,-

I

;_i

line

profile with

the

of this

Ejection

mechanisms

in the

sublaver

41 It

¥

FIGURE away

5.

Three

from

mately y-axis

the

dimensional

viscous

25% higher corresponds

is not clear,

are

to some

In fact,

when

mensional that

structures periods, course

much

velocity

extending

hot

spots

of the

wall

they

(figure

of their

6, where

life they

several time

a structure

runs

has

next

from

began

frame

vorticity

appears

boundary

layer

reproduce,

of these

at and

has

the

wall.

fuses

with

Finally the

first

high,

the

tick

showing

that

form

of leaning

to follow

An

In the

first

the

a small

the

tip of the

vortex

and at

tip separates from its parent structure, forming of a new spot. The last frame shows both spots

for fairly long widths. In the

walls

at

grows

sequence

its

patch

same

These

structure

regions in figure

of reverse 3. The

vorticity

whole

are

reproduction

visible

top

process

of the instability process for 2-D linear Tollmien-Schlichting Criminale, 1967). Basically, vorticity is created at the main fluid

flow through below

the

viscosity.

critical

layer

In a frame is moving

of reference backwards,

is strikingly wall

moving while

lateral

that

end.

of strong

out

into

moment,

to be the independent

in the

able

in figure

time

At this

what appears as essentially

we were

is given blob

layer.

di-

of these

a small

the

of three

the the

"embryo" units.

A closer examination of the vorticity field shows that there is a region centrated z-vorticity of opposite sign (negative), underneath the top part structure.

spots

evolution

of this

vorticity

stretching

the

"necks"

and

example

frame

considerably,

in the

curving

to new structures,

processes.

mark

sublayer.

flow in the

a "forest"

origin

producing grown

the

is approxi-

some of them were followed to move several channel half

to bottom.

blob

into

extending

represented

viscous

5). It is possible

giving

to stretch,

the

the

form

reproduction top

and

is quite

outside

are followed

of z-vorticity,

structures

iso-surface

wall vorticity,

as they move in time, and long enough for the structure

to observe

In the

these

of z-vorticity

vorticity

this convection

structure

iso-surfaces

covers

The

than the average to y+ = 17.

agreement linked

representation

sublayer.

of conof the

view

of this

reminiscent

waves (see Betchov & and diffuses into the with

the

on top

structure, moves

the

forward.

42

Jimenez,

Moin,

Moser

and Keefe X

¥

P'f

(

x

((

xx

(d) FIGURE 6. explanation.

z

Splitting process of a structure Time difference between frames

to create a new one. See text for is approximately 6 wall units (t,/u_).

Ejection As a result, ing

"V"

main

as the

shape

vorticity

that

structure

backwards

the

on the The

the

wall

of the that

original

vorticity

in figure

critical

positive original

3.

the

induces

part

the

and layer

begins

of the

vortex

carries

structure.

vortex

a new

in this

the layer

part

of the

The

negative

the

positive

fashion,

wall,

is convected

overcomes

to induce

in the

near sign

a negative

eventually

point-

vorticity

of opposite

forms

in formed that

a backwards

positive

a negative

outward,

pair

43 it takes

vorticity

This new

is created

upper

wall,

Eventually,

and

vortex

the

This

structure,

an updraft into

from

layer)

one.

a strong

structure,

away

as it diffuses

original

moment

ative

the

in the sublayer

no slip condition.

by the flow,

derneath

diffuses

is visible

(above

to accommodate

mechanisms

uneffect

vortex.

underneath positive

the

and

neg-

vorticity

in the

rear cuts the connection between viscous annihilation), while the

the head and the base of old structure (through positive vorticity connects with the head of the

old

(see figure

structure

to form

a new

here, for the production ejection of shear layers As

noted

waves, splitting

of the updraft, into the flow.

previously,

and

one

this

it seems

is the

to explain

mechanism

in figure

of ejections

in the

undoubtedly

some

sublayer three

(figure

corridors, it. seems Also,

8).

The

as a necklace to be weaker,

there

is little

6. This may

stream, longitudinal evolution. The

general

lection "ramps" lines

picture

of patches cannot

end

fluctuations

bounding wider experiments.

one

associated

and

shear

three

correspond if the

layers

in the

layers

to the like

the

these of high

velocity.

a unique

opportunity

natural

channels

present

in the flow,

a physical

is complicated

experiment.

and

since The

to attempt

this,

numerical

the

among

those

sublayer, studied

but here.

into

are

the

induced

the

the

to the

observed

velocity

is a colinto

Since

linked

the

the main in their

model

of high

since

to ride

important

in this paper.

is precisely

difficult

equivalent

seem

lifted

9, and

by the interactions it is obviously

the

simplified

The remaining question is whether the three dimensional of the protruding ramps can be studied in isolation.

provides

that

to be in between

consist This

by the fact

a map of y-vorticity "corridors" between

are

V-vorticity

in the are

further

are

pattern

little vortex

wall

8. the

longitudinal streaks,

observed

ill

representing simulation

behavior

of structures

the large

number

of isolating

by

in figure streaks,

narrow

structure Numerical

to isolate

T-S

there

which

ramps

one in figure should

layer,

described

are constrained

sublayer

of lower

regions

by this

2-D

for the generation

present in the the structures effects

wall

linear

although

layers

are ejected

dimensional

of the flow,

ramps

something bands

shear

nfiddle

shear

x-vorticity is also to correlate with

in the

which

the mechanism as shown

suggested

in the

the

2 for the

of z-vorticity

present,

vorticity

to see that, becomes

their

as the

to the

behavior

invoked

in figure

for

dimensional,

sublayer

"sidewalls"

velocity

effects

mechanism

proposed

two

of the

It is easy picture

that

of the high

corresponding

the

into spanwise bands. In fact streaks, delimiting quiescent

of beads, and harder

vorticity

one

the

suggests

and

that,

that

responsible

be essentially

spots,

doubt

same

mechanism

dimensional

hot

is the

Note

approximately

structures do not spread laterally in the sublayer shows long active them

7).

in

of them

a single

structure

in

a single

structure

is

44

Jimenez,

Moin,

Moser

and

Keefe

t

FIGURE

2-D

7.

Tentative

mechanism

Schlichting

waves.

mechanism

corresponds

roughly

for to

the the

splitting instability

process

of a structure.

mechanism

for

This

Tollmien-

Ejection

FIGURE are

8.

Streamwise

identical

that

the

to those

vorticity

velocity

streak

pattern

of figure

4, and

spots

reside

in the

in the subla!ter

of 9-vorticity careful

45

in the

comparison

relatively

quiescent

sublayer.

with

Conditions

that

corridors

figure

between

shows the

high

streaks.

FIGURE

9.

portions

of wall

A model

of the

to use

a computational

or at most Running 2_rh × 2_rh the

for the

vorticity,

origin

from

mechanisms

9-vorticity

sublayer

and

are

box

whose

z-z

This

at Re = 7000,

we first

(about

As expected,

limit the

1800

× 1800

cycle

tried

wall

solution,

flow became

extent

for a long

time,

previously

available

out

computational

using

to approximately

flow

raised

structures sidewalls

are that

quickly

it was

fields.

The

boxes units.

both z and z, this corresponds spacing close to the one found

initial

addition

not

any

lifted

are

the

whose Since

the

were 3-D

the

to understand

numerical

spanwise

on a periodic

conditions and

only

one,

workshop. box of extracted

perturbation.

wall

shear

stress

2-D (Herbert) solution (w. = 3), (_o: = 12). This solution was no(.

easier

remaining

of the

of a small

three-dimensional, to the channel

to contain

course

a channel

The

the

enough

in the

computing

units).

with

since

113 wall

is small

was attempted

grew from the low value corresponding to that for a fully developed turbulent followed

The

by vorticity

streaks.

a few, structures.

2-D

structures.

"supported"

extent

computational

to a periodic array in natural flows.

than

experiments was

7rh/8, domain

any were

of the carried

corresponding is periodic

of structures with We tried different

a spanwise streamwise

in

46

Jirnenez,

Moin,

Moser

and

Keefe

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FIGURE

10.

Note

that

and

smooth,

z-vorticity

the

none

the

short

very

and

most

thai

sufficiently

while

by the

contains

dimensional

in which other

one

solution

fairly typical, characteristic

computationa]

at

model

for its study.

of the

(the

top

one

top

wall

(see

layer,

but

Further

case) 10).

at this

moment,

constitute,

with

a possible

model

Apparently,

boxes

a roughly

layer, This

presumably

boundary and

are

nor

is a non-

sustains

at the wall falls At each particular which

it is

solution,

result

layers,

structures,

experiments

The

of

since

to a 2-D

boundary

figure

to laminar

an z-extent

wavelength.

in our

in 3-D natural

3 or 4 sublayer

with

one.

3-D

(actually

to lanfinar,

to decay

turbulent

a turbulent, to those

decay

were

perturbations

decayed

one

of this

average shear stress and 3-D solutions.

box contains

available

the

similar

but the of 2-D

a simple

wall has

3-D

boxes

structure

quickly been

cannot

stable

desirable

to introduce

has

text.

"turbulent".

a single

27rh/5

perturbations sufficiently

one

structures

and flow

a fully

most

attempts

to now,

This

to maintain

2-D

ejection

up

The

in the

two-dimensional,

and

to contain

of rrh/8,

of rch/8.

as described

three-dimensional

However,

_._,.'.'"":'._

is essentially

successful.

box,

..... :;-L " .¢">

channel,

figure,

be expected

is neither

the

of the

completely

periods

to two

narrow

remains

could

one

domain

channel

flow,

one

interesting

unstable

symmetric

top

of them).

streamwise

unstable

forced

that array

a spanwise

however,

to be values

was

The

linearly

2-D

of which

with

solutions. 2rrh,

lower

ones,

of a 3-D

at. the

the

periodic

in boxes

wall,

while

periods, a doubly

upper

field

:

:- z_:Y':;:.-'.:.".-.:,-_"." :,> -"

layer

which

appear

in between moment,

still

the

too

many

smallest

of different

sizes

the the for

system are

still

in

progress. Summary In summary, tions that

in the this

vorticity, which

we present viscous

part and

have

Schlichting

waves.

study

have

failed

sence

of external

in the

form

sublayer

of the

is essentially

here

flow

of a turbulent

proposed The

to that

up to now,

of a long

for their

it appears

that

GF:CI:TAL OF IGOR

of vorticity

channel.

We

strong

single

and

computational

P.XGE QUALrl_

IS

structures box

ejecshown

layers

of z-

reproduction

of 2-D Tollmien-

a single model

have

shear

maintenance

computational

periodic

inception

for the instability

computationally

a convenient

narrow

protruding

responsible

to isolate since

but and

by

a mechanism

efforts

forcing,

rectangular

is dominated

equivalent

for the

structure decay

has

been

containing

for its in the

ab-

identified at

each

Ejection moment

only

systems

is in progress.

mechanisms

a few structures.

in the

Further

work

sublayer

in the

47

identification

of better

reduced

REFERENCES BETCI-IOV, R. & CRIMINALE, Ch. II. CANTWELL, 13,

T.

1976

Con].

Num.

eds.).

Springer,

JIMENEZ,

J.

Phys. J.

1987

J.,

The

Periodic

Stability

motion

of parallel

in turbulent

flows.

flow.

Ann.

Academic

Press,

Rev.

Mech.

Fluid

in Fluid

Bifurcations

flow.

Dyn.

and

Turb.

(A.I.

bursting

P.

of a vortical Shear

in a plane Van

channel

de Vooren

. Proc.

8z P.J.

in two-dimensional

_5 MOSER,

S.J., REYNOLDS,

R.

associated

6, Springer, 1987

number.

W.C.,

of turbulent

structure

Flow

flow at low Reynolds

structure

motions

5th lnt.

Zandbergen,

Poiseuille

flow.

bursting

event

in press.

Evolution

MOIN,

secondary

pp.235-240.

1987

Fluids,

channel KLINE,

Organised

Methods

in channel KIM,

1981

1967

457-515.

HERBERT,

KIM,

B.J.

W.

Turbulence J.Fluid

SCHRAUB,

boundary

pp.

layers.

with

statistics Mech.

F.A.

&

J.Fluid

the

221-233.

177,

in fully

RUNSTADLER, Mech.

developed

133-166.

30,

P.W. 741-773.

1967

Center

for

Turbulence

Proceedings

of the

Program

Overview The projects with a common

49

Research

Summer

of

of the theme

1987

the

K-space

Group

K-space group consisted of the interaction among

Projects

of six independent lines of inquiry scales of motion in turbulence. The

studies were almost entirely limited to homogeneous turbulence. paragraphs I will give a brief overview of the six projects. The invited

participants

were:

Jean-Pierre

Bertoglio

(Ecole

Julian

Hunt

Paolo

Orlandi

Roland local

Buell

Gary

Alan

Wray

the

at

which been

to test level

is the

they

collapse

the

at the

EDQNM

bulence tures.

An

experimental

inject

known

numerical for this

two-time

higher

wave

coherent

flow

coherency.

structures

conditions

for such

passing

frequency

into

illuminate

the

of the

been

energy

propellers

while

that

Orlandi's

project

for the

energy

•

,.,

_,

concerns

the

simulation.

in the that

transfer

,

between

'

4 •,

,•-

the

•

,

such

computationaly

,_

-

important such

experiment

models resolved

theory has

to define

to solve that to do so.

peaked

(or supergrid) and

accurately unresolved

with initial

at the

mystery,

must

to

a parallel and

theory

peaked

as a subgrid

struc-

propellers

to conduct

statistical

tur-

structures.

with

using

attempted

has

of a joint

coherent

simulation

in the

use of EDQNM

At a minimum,

the

of

collapse

simulation,

Bertoglio

to

at Stanford.

many

and

experiment,

rotation frequency. It should be a simple matter was simply not enough time during the workshop for a large-eddy

Ferziger

in France flow,

to be a

simulations

good

copes

in

degree

continuation

EDQNM

a turbulent

difficulties

The

and

whereas

started

step

by the

in direct

and

EDQNM

is assumed

of persistent

well

workshop

a sinmlation.

which

is the

Squires

between

any

During

work

the

important

functionals

in nature, how

has

scale

promising,

by the presence

Comparisons

should

imbedded

and

statistical

program

simulation.

very This

in France,

is to determine

An

auto-correlations

appears

dominated

goal

within

done.

candidate

numbers.

is purely

assumptions

time

several

velocity

approach

are

second

previously

been

judges

co-workers

problems

of the

has

This

theory

Bertoglio's

accuracy

of a Lagrangian

Bertoglio

by Bertoglio,

The

the than

estimation

turbulence.

achieved

effort

Ames)

wishes

of E(k).

isotropic

Ames)

a deeper

closure

functional

University)

(NASA

(NASA

Bertoglio closure

University)

(Stanford

Rogallo

de la Turbulence)

Ames)

(Stanford

Ferziger

Robert

Statistique

of Tokyo)

were:

(NASA

Coleman

Joel

de Mecanique

(University

participants

Jeffrey

de Lyon)

of Rome)

(Institut

Yoshizawa

Centrale

following

of Cambridge)

(University

Schiestel

Akira The

(University

In the

bladeat their

but

there model

account scales.

5O In EDQNM,

the

interacting demonstrated

of isotropic

and

also

spectrum

of the

if the

for their

effect

of the

The viscosity

Chollet,

and

a gradient

Lesieur)

Another but

but

the

in the

resolution

increases, than

uniform number

the case

clues

some

The time rian

axes mean

project

frequency

that

and

both

be extracted of the stress

Buell,

Ferziger

(which

tractable

requires

additional

because

The are,

in

of this, the theory are however very

consider

calculation.

as the grid much

more

the possibility

angular

in the

and and

not

or three

covered

of

distribution

proper

variables,

strain

the

Wray their

EDQNM concerns

wanted

influences

results

support

way:

tensors,

appear the

In any from and

to

choice

of

expansion

is crucial.

theory,

In this

consideration

from

the manner

equations. the

dependence

they

There

independent,

rate

of

of homogeneous sheared authors speculated that

variables.

and

distribution

functions.

rapid-distortion

mean

angular

in a general

dependent

ones The

the

of the dependent

enters

). In particular, number.

and

of two

from and

by large-strong

wave

to ho-

calculation

homogeneity

in a direct simulation quite smooth and the

of variables,

(or correlations), scales

closure

and

less tractable,

by inspecting

were

flow gradient

or Lagrangian

small-weak

tensor were

choice

of Hunt,

spectra

as was

of Siggia,

in the

accurate

approaches

shells

required

the

might

principal the

and

scales, works

in

be represented by the weighted sum of a small Coleman and Ferziger attempted to estimate

by a sum

issues

system

but

physically

is uniform

EDQNM

problems

for LES calculations

EDQNM

spherical

functions

important the choice

in which

over

flow) would basis functions.

technique of the

it is far

Coleman

the

stress spectrum The distributions

coordinate

the

to be reasonably

models

isotropy.

be represented

be several

that

( see the

with

of forcing seem

at the large

accuracy

contribution

to simplify

of basis

could

strain

simulations to extend

flows

subgrid

correlations

the Reynolds turbulence.

appears

of subgrid

the

in isotropic of smooth

number

they

context

approach

of velocity

of a mean

not

form

of Kraichnan,

sort

does

acthe

momentum

diffusion works

some

viscosity,

can

however,

in the

(see the

not well understood

in anisotropic

it approaches

a Galerkin

terms

is presumably

numerical

EDQNM

simulation

by a gradient

eddy

is considis considered.

be estimated, In the

over

Orlandi has in a direct

of interactions

is much more expensive to compute. Because much attention for anisotropic flows. These

important rapidly

integral

required.

model

turbulence,

assumptions, and has not received

term

attempted

integrals

set

transfer

application

encountered

EDQNM

subgrid

art of driving

also

can

modeled

the

is currently

). Orlandi

five-dimensional

isotropic

as the

of interactions

as additional

negative

The

full

transfer.

appear

is the

in time.

etc.

scales

supergrid with

possibility

shear,

While

the

. The

the

spectrum)

is usually from

et al above,

Pope,

mogeneous of the

k is calculated

when

energy

must

term

form,

random

by Hunt

to the

determined

diffusion

both (truncated

scales

subgrid

an eddy

Kerr,

number

unresolved

contribution

unresolved

equations.

done

wave

turbulence a subset

count

correct.

into

when

Then,

space

transfer

triads (k,p,q) of a functional of E(k), E(p), and E(q). that EDQNM accurately produces the transfer measured

simulation ered,

net

relation

on the

to determine the

between

reference how

Eulerian the

assertion

time

space

frame the

advection

spectra of Tennekes

and

( Euleof

at high that

51 such

advection

that

Hunt's

was

over

p.d.f,

with

these

earlier

the

small

In

model,

in wave

the

information

that

space,

closures shells.

hoped

to determine

some

clues

were

to aid

modeled

enough were

rather

noisy

at the

elsewhere,

well

and

past several Navier-Stokes

weak.

scales

mally

averaging

large

scales

features

are over

field

scale,

(local

to homogeneous

to its isotropy. DIA

framework

of these

using

simulation

could

not

the

database

available for the

prior

small

to find the

to the

determination

entire

As a result,

some

quantities in addition, and do not

and

program that

of several

and

to get

of the

terms

there the

sample

was

order, small scales form

not

simulations

and

are simply

and

Taylor

are

sometimes

not

available

series

and

the

scales

upon

scales

(formally,

scale

the the hope

contribution

of the

Unfortunately had

during

was

available

constants.

diffusion

functions

to be able

not

sample

in

to test

higher-order

program

was The

gauge

this expan-

the

been

the

leads by the

Within

some

not

small-

disparity

asymptotic

Yoshizawa's

to be

, at small

modeled

for example

by replacing

and

of the

anisotropic.

closures,

of the

deterministic

leads

are in turn

become

statistics

expansion

features

time

by for-

are assumed

depend

spatial

expanscales is

separated

scales

statistical

of the

postulated

data

some from

small

in general

in one-point

required

using,

taken

of the small

the constants.

the

the

unfortunately,

time,

at which

the

summer

is

needs.

and

to estimate the

over

worked out a formal two-scale the interaction between the

It was

because

num-

tensor

the same

expect,

But they

Schiestel

constants".

to do so. In addition, simultaneous

of the

terms

the

data to determine

be computed

because

of the

is then

and

and,

statistics

disparity

be modeled

"model

models

not,

The

at lowest

order

A model

with

would

averages

statistics

it is possible must

energy.

expansion

several

At higher that

As one

(its derivatives)

turbulence required

(TSDIA)

of terms

of kinetic the

The

model

he is currently

scale and

The

than

formulations.

one must

space

interaction

field

correlations).

formalism.

sion)

The

data; modes,

low Reynolds

spectrum

models

what

in both

to be valid

converged.

rather

anisotropy.

an intermediate

of the large-scale

scale

scales to mesh

disparate

computed

dissipation, etc.) and are globally conservative

years, Yoshizawa has equations in which

is assumed

statistically

due

agrees

of forced

and the

shells

stress

were

to be precisely

Over the sion of the The

larger

scales

appear

others

improvements.

at the

small

a more

which

Schiestel's goal during the summer program terms with data from direct simulations. He

of the

while

in the sample

stress

each shell,

improvement.

to consider

using

relation

Reynolds

( pressure-strain, These transfers

accuracy

in their

rather

time

biased

the

spectrum

resolution.

Reynolds

appear in the one-point approaches. was to compare the models for these

analysis

observed

over spherical

Within

the

a simple

be retained,

but indicates

to the space

statistical

for the

one-point

is retained.

were small

could

frequency,

spectrum

reworked

numerical

equation

number

as in classical one-point model transfers between

Hunt

of the

scales

at high

and derived

anomalies

for adequate

as done in classical

scale

project velocity

Some

of spatial

required

Schiestel's

space

the

simulation.

was

integrated,

During

spectrum

the time-space

to be a consequence

range

ber that

relating

for the advecting

appear

the

the frequency

proposal

simplified.

realistic well

dominates

terms,

model

terms

extracted

from

no

really

one

was

adequate

was too

small,

52 as was terms

the

Reynolds

suggested

one notable example,

exception

scalar

gradient

was

having itself

the

case

a mean

close

reduces

gradient time

diffusion

in the treated

to omit

to the values

the

held

case

case

is appropriate

for testing

All of the

projects

are,

in my opinion,

worthwhile

pursued

further.

of the

data

previously

fixed

predicted.

In this

For

predicted.

shear

in the

new

well ( with

rate was correctly

direction.

which

most

fit the

in homogeneous

stream

but was in fact

simulation

forced

the models

the dissipation

of turbulent

with

A later

clear

was then

this was done

constants

rotation

changes

by Yoshizawa. it is not

Yoshizawa When

) with

the fact that

The exception sive

number.

by TSDIA.

case

of a pasthe

mean

simulation

of changing

mean

Yoshizawa's

model.

used

gradient

but

Summary

be

by the

number

puters.

The

Evgeny

Novikov,

We at exposed

Ames

The

of local

group

progress

benefited

and

several

benefited

from

to areas

made

participants

of research

during

available

and the

feasible,

summer

to interact

greatly

from

projects

were

directly

influenced

of these

projects

because

each and

ideas

that

discussions

with

and

will hopefully

session

directly Robert

was

with

com-

Kraichnan

and

by their in every

we had not pursued

limited

the

suggestions. case

in-house Bob

we were before. Rogallo

N88-23091 Center

]or

Turbulence

Proceedings

of the

53

Research

Summer

Program

1987

EDQNM closure: "A homogeneous simulation to support it", "A quasi-homogeneous simulation to disprove By

It is known turbulence. vian

J.P.

(EDQNM)

approach

direct

simulations

the validity

of two-point

DNS

to validate,

or improve

In the lected

first

part of this

a case for which

isotropic

turbulence.

assumptions

The

work

EDQNM

of two-point

aim of the second

Our purpose

is to build

results.

present

We

ditions.

structures based

and

Lagrangian

two-point about

damping triple

theory

of a damping coefficient

among

tt(k, p, q) is the sum p(k,p,q)

1

Ecole

2

Stanford

Centrale

de

University

Lyon

can be

of the present

results: may

work

EDQNM

injecting

is likely

conditions is simple time

to dissituation.

to give

in the

initial

accurately

scales.

turbulent

of field

initial are

poor field con-

expected

enough

contains

accounted

the

improve)

difficult

them

but at the same

time

simulation

more

a quasi-homogeneous

to be

to test

possibly

homogeneous

we se-

homogeneous

be used

(and

in a much

such

it")

Lagrangian

to validate

this kind

times

in isotropic

The Assumptions

the

growth

term

to al-

"coherent"

for by

closures

the

three

of the

third

time

order

moments

equation

for the

an inverse

time

wave-vectors

inverse

turbulence of EDQNM

of the

in the rate

p(k, p, q) is essentially

correlation

useful.

that

Gaussianity.

spectral

EDQNM

study

using

theories,

be expected

most

to support

spectral

by artificially since

time,

purpose

which

case for which

simulations

same

to give satisfactory

to generate

for EDQNM

1.I In the

2

information

simulation

of EDQNM

a reference

cannot

on expansions

a test

of direct

with

It is the

quantities

refined

structures,

a challenge which

introduction

Ferziger

EDQNM.

approximations:

an attempt

The results

low comparisons

that

is known

at the

part of the work ( "a quasi

"organized"

to provide

J.H.

can provide

theories. upon,

closure

a careful

and,

(DNS)

evaluated

it") is, on the contrary,

containing

2, and

("a homogeneous

We then

Our goal is to make the EDQNM theory. prove

Squires

is one of the simplest

numerical

to test

to use

1.

1, K.

that two-point closures are useful tools for understanding and predicting Among the various closures, the Eddy Damped Quasi-Normal Marko-

Nowadays, used

Bertoglio

it"

of a triad. scales

for the

is limited

triple scale

by the

correlations. for the

decay

It is generally individual

+ r,(k2 + p2 + q2) = _(k) + 7/(p)+ r/(q)

The of the

assumed

wavenumbers:

54

J.P.

and

r/(k)

For the

is specified

flows

Bertoglio,

and

numbers

coefficient

must

sufficiently

have

the

form

form

proposed

by Pouquet

high

to contain 1970)

commonly

According

to this

wavenumbers There

scales

patible

with

introduced

the

RNG

stay

only

Indeed,

approach

of Yakhot

type

for Turbulence

the

rl(k ) = A,[

where

v> (k) is given

,Xa and

satisfy

the

Sb are constants.

following

During

that

the

no significant

it into

figure

1); the other

form

small

be completely (CTR)

1987 a time

+ (v + v>(k))k

fk

damping).

the

effect

of the

closure

com-

(1987)

recently

scales

affect

correct,

and

summer

program

scale

that

the dur-

depends

2

E(p)dp ,7(k) +

It can be shown

reduces

Summer the

differences

as )_a/)_ remains

the

on

that

these

two new constants

must

relation:

new

the CTR

introducing

the

only

by :

)t, = )_ -

in order

that

a two-point

of building

p2E(p)dp]½

= where

can

possibility

k depends

Kraichnan

only

Research

to be 0.355.

influence

Orszag,

that

of expression

following

eddies

to devise

and

it is assumed

taken

to assume

in order

Center

Kraichnan suggested on all scales:

large

spectrum:

+ vk 2

at wavenumber

grounds,

neither

at the

the

_t

energy

)_ is generally

scale

on physical

in which

for an arbitrary

p 2 E(p)dp]2

k

time

k (i.e.

reason,

subrange,

2

of constant

the

can be neglected.

In fact,

his

value

than

no

a model

damping.

The

expression

smaller

is however

smaller

ing

used.

an inertial

= a e_ks

et a1(1975)

rl(k ) = )t[ f0

is more

Ferziger

(Orszag 1

_(k)

The

J.H.

phenomenologically.

at Reynolds

damping

K. Squires,

larger

from than

quantities

to the

Program,

EDQNM

2(1 - log2)_ -_

old one in the

we tested

computation the

0.5.

results

obtained

We noticed

remain

the code

unaffected.

new

case form

written using

a small

of an inertial

range.

for the damping

by Orlandi.

by

We found

the

classical

form,

effect

on the

skewness,

as long (see

EDQNM

closure:

Two

55

views

.6

.4 -$

.2 In)

0 •6

i

,

i

I

i .5

i 1.0

i 1.5

! 2.0

I

I

"

.4 -$

.2 (b) 0

I 2.5

I 3.0

t

FIGURE

1.

Evolution

obtained

with different

et al (1975);

(b)

of the

formulations

damping

In order

we have

Comparisons proper

the time

since

the

scale

must

time

scale

triple

must

correlations,

be

invariant must

introduced

by

Ui(X,

through

x at

Kraichnan but

t'lt)

is the velocity time

in the

the evaluation Spectral

initial

One

actual

the

turbulence

simulation

results.

Approximation of the

by

.5).

against

arbitrary

velocity

Galilean

convective

derived

For practical

from

suggest field.

transformation effects.

Tile

Lagrangian

reasons,

The

we used

time

two-time the

two-time

=

at time could

t of the fluid particle

also

or ALHDIA

of Kaneda's

information

position

To deduce

t'. LHDIA

scale

by Pouquet

(1981):

Rij(x,t'lt;x',t'lt') where

an

:

from

Interaction

be affected time

LHDIA.

by Kaneda

scale

given

results

simulation.

in EDQNM

correlations

under

not

(a) damping

numerical

the Direct

of EDQNM

(with)_a/_

time

two-time

be a Lagrangian

as suggested

correlations

and

using

correlations

therefore

a spectral

EDQNM scale

direct

for r/(k) used

to deduce

between

evaluating

Comparison

by Kraichnan

of 7l( k ) using

to test the expression

scales,

in time.

for the damping.

suggested

1.2 Evaluation

time

skewness

use

correlations

is obtained

the

theories,

whose

two-time or work

trajectory

correlations with

the

passes

defined

response

by

tensor,

is simpler.

by

Fourier

correlations

from

transforming

with

respect

to

the

of the particles. the

Lagrangian

the

results

of a DNS,

one

has

to

56

J.P.

follow

particle

for the

Bertoglio,

trajectories.

study

K. Squires,

This

of particle

is done

and

using

J.H.

Ferziger

Squires'

code,

which

was

developed

dispersion. 1.3 Results

Two Both

direct used

numerical

Rogallo's

a 128x128x128

grid

We present the

paper

this

introduced

only

DNS,

in Lyon have

computational

grid,

doing

prohibitively.

The

large

points.

scale

a set set,

In are

markers

of widely

2a, the

Lagrangian

time

curve,

figure

scale;

at

high

nan's

expression

that, (with

remains

The bers.

major The

conclusive

results

In the lem

problem

number

decreases

faster

Fourier

than

is sufficient

on the

rotating case

The

to uniform frame) during

the

about

solid can

by the to a single

the

collapse

When

is

Kraich-

improved

(figure

at low

2c).

at high

large

initial

positions

k, the

another

Eulerian

time,

the the

of the

from

the

prob-

correlation trend

appears

significance

markers

for the spectral

deduced

to allow

simulation.

for long times,

definition

wavenum-

enough

correlations,

separation

time

small

of

must scale

time

be

in this

behavior

study. the

present

the effects

be used.

summer

results

is used,

in the

of the

to the

anisotropic

body

the

not

at high

one; however,

of our

easiest

the

normalized

sample

Eulerian

and

information

to extend

is available

damping.

submitted

purposes

be interesting

no information

that

run

As expected,

is slightly

of sufficient

and

to use another

we believe

collapse

was probably

times

values

respect

try

time

scale

wavenumbers

Lagrangian for short

with

for the

It would

highest

, for large

One could

However,

(8x8x64)

the Lagrangian

In fact

transforming

questioned.

this

between

As expected,

to be reversed.

the

the

comparison

appears.

case.

concerning

a

rapidly.

wavenumbers

is tim lack

used

This

scales,

128x128x128

t'.

at low wavenumbers.

the

grid.

small

and futuT_c orientations

encountered

of particles

time

the t -

collapse

should

at high

1._ Problems

from time

scale

k,a = A/2),

number accurate

used.

of the

classical

the

spaced the

of the

computation

than

on

decrease

as good

to grid.

point

to ensure

equally

as functions

the

acceptable

smaller First,

obtained

wavenumbers

not

cost

data

separation

time

but

is used and

was

when

wavenumbers

wavenumbers,

of 64 particles

be plotted

an accurate

2b shows

good

obtain

dimensional

also

of the

To

to large

can

the

on a 16x16x16

see

on a 16x16x16

increased much

study

preliminary

at each

correlations

of the

corresponding

correlations

candidate

placed

detailed

A study

types.

used.

second,

number.

particles

of two

particles.

of 8x8 lines

as functions

correlations

were

spaced

Reynolds

Simulation

is therefore were

In the

to place

have

of markers

the

(1987).

Eddy

were

was used.

For a more

Ferziger

Large

so would

placements

consisting

figure given

The

number

Marker

statistics,

provides second

but

to increase

desirable

turbulence

grid

of results.

and

using

been

isotropic

a 64x64x64

number

Bertoglio,

it would

of grid

case, in order

a limited

Squires,

was performed

In the

of homogeneous

In the first

was

here

by Lee,

work

simulations

code.

Due

of mean case

rotation

program.

study

lack

It should

velocity

in this case, of time, be the

flows,

as almost

gradients

or anisotropy

would

be turbulence

to investigate

since,

to the

to anisotropic

steady we could

subject

coordinates not

of future

investigate work.

(in

EDQNM

cloJure:

Two

view8

57

1.0 .8

_0 _.4 .2

10-3

1.0 .8

_0 _.4 .2 0 10-2

10-1

100

DIMENSIONLESS TIME 1.0

---

._.:_..

.

.8 "',, _..?_

\ +j_

_4

k - 1"

,,

_

\-\\\\

',,,,,__.,

.2

(_!...

, ,,,,

10-2

FIGURE

2.

tion

at different

time

separation

using

Two-time

Kraichnan's

values

Lagrangian

correlations

of the wavenumber.

normalized formula.

using

,

......

10-1 DIMENSIONLESS TIME

Pouquet's

'_ 100

as a function

(a) un-normalized formula,

(c) time

of the time

time

separa-

separation,

separation

(b)

normalized

58

J.P.

In the ited,

case

since

of isotropic

it is known

of the

decay

reason

why

Bcrtoglio, fields,

that

of isotropic our

goal

provide

insight

formulations This

Since of the the

information of the

term

is quite

to satisfactory

is, to a certain

extent,

turbulence).

lim-

predictions

true,

(and

However

the results turbulence

the accuracy

effect

damping

lead

is the

we believe

of the present study containing a gap or

of the expression

the choice of the triadic interaction

interesting

Ferziger

the

of anisotropic

for example,

about

J.H.

to isotropic turbulence, of homogeneous isotropic

is of greater importance. effect on the "localness" possibly,

existing

study

and

in studying

turbulence.

is the

in the spectrum,

for r/ could

interest

the

that, even though limited are valuable. In the study a peak

K. Squires,

for the

damping

damping term has an important in k space, a better expression

concerning presence

localized

of coherent

interactions

structures

ergy cascade (see Kraichnan, 1987). Furthermore, we believe simulation, the influence of the subgrid terms on the Lagrangian

and,

on the

en-

that, in large-eddy time scales should

be investigated. 2.

A quasi-homogeneous

field

2._ Aim The first is known

part

to perform

significantly simple

of tile

likely

to

to provide

of the study

work

was

well:

different.

enough

containing

It consists

be handled a severe

initial

turbulence. closures

would,

The

aim

a case

reasonable

into

riodic

into the structures

however,

ability are

require

During of a flow such

a "quasi-homogeneous"

rather

than

the

summer

a simulation

Lyon. by

using

figure

3.

could

a large

Large

are

results

a spike

containing tendencies existence

such

consists

equipped

Eddy

we tried

eddies? bounded

flows

amount

field

is

although times,

is well

known.

of computational have inject

is

effort

for of Wall to be

to be studied. coherent struc-

(quasi-homogeneous

reference

small

in this

(see

Michard

appears

data

meaning

in the on

rotating al,

(one 1986)

pe-

distribution

run

for a simulation

box.

The results

of

EDQNM. has

structures One

at each

display

see figure

grids,

four vortices embedded in a random can be predicted, and, to a certain of a periodic

cubic

propellers.

spectrum;

16x16x16

conditions

an experiment

organized

experiment et

to test

field,

of generating

with

initial

in a periodic

a turbulent

used

Simulations

to define

structures

provide

to create

a grid

particular,

the

program

propellers The

which,

part

of two-point closures to account present. How are the predictions

turbulent

"coherent"

This experiment

rotating

second

homogeneous).

containing

In an attempt

EDQNM

computation

predicted by EDQNM, and at least at first, simpler situations The basic idea of the present work is therefore to artificially tures

in which

of the

to simulate

with

EDQNM affected by the presence of organized The existence of coherent structures in wall flows

of a case

of EDQNM.

Our goal is to get insight situations in which coherent

bounded

structures

conditions.

to the study

of an attempt by

test

and

devoted

isotropic

"organized"

been

devised

in

in grid turbulence hundred

node

of the

interesting

countergrid);

see

behavior,

in

4.

in Lyon,

using

initial

conditions

field, show that some experimental extent, understood (for example,

of anisotropy).

However,

due

to the

lack

E D Q N M closure: Two views

59

FIGURE 3. Grid equipped with propellers (Michard et a1,1986).

Hz

F I G U R4.E Experinient.al one-dimensional energy spect,rumof Michard et a1 (1986).

of resolution of this coarse grid, it has not been possible to inject a peak in the spectrum in the flow direction. The work done at CTR consists in building an initial field that contains such a peak. We used a 64x64~64mesh. A few simulation runs were started to test the initial field. After several attempts, it was decided to define the initial conditions in terms of the vorticity field. In the 64x64~64box, four vortex-containing cells, corresponding to the wakes of four propellers, were defined. In each cell three vortices are introduced, centered on helical lines corresponding to the wakes of the three blades of each propeller. In order to ensure zero circulation in each cell, another vortex was introduced on the centerline. A representation of this initial

field is given in figure 5. This field was embedded in a random fluctuating field. For this random field, we

60

J.P. Bertoglio, K . Squares, a n d J.H. Ferziger

FIGURE 5.

Representation of the initial “coherent structures” used in the simulation. These are t,he idealized wakes of the counter-rotating propellers shown in figure 3.

.03-

FIGURE 6.

+

Initial one dimensional spectra (vortex field isot,ropic turbulence, averaged over the entire 21 - 23 space). The mean flow is in the 22 direction. Ell, - - - - E227 - - E 3 3 *

used the one created by Lee & Reynolds (1985) with N = 64, and Re = 51.13.

EDQNM The

initial

appears

spectrum,

at the

in the

wavenumber

The

peak

must and

in figure

field

one

out

that

point

It is worth

evolved The

noting

that

only

stream

longer grid.

was

in the

This

existence

the

It could

of a reference

able since precise deduce from the

passing

of the peak

deduced

from

the

steps

at

(due

to

is found

at

peak

frequency. due

to

time

behavior. contraction

introduced

that

down-

to amplify in fact,

this work

simulation

am-

should

will be completed

as preliminary

is overpredicted

the

the

contraction

for EDQNM,

occurring

obtained

rotation

time

it

plane,

computed

die immediately;

the

of the mechanism quantities.

X,Z

spike

The axisymmetric

decay

the

frequency

at the

was

It is hoped

in the

However,

spectra

hundred

would

expected. a challenge

field

analysis measured

over

is an axisymmetric

provide

experimental

contained field.

experimental

appeared

contraction

in the simulation.

future.

that

there

it, the peak than

blade

to see the expected

axisymmetric

stronger

energy of the

the simulation,

spike

peak

work. the

the

to one

is that

Without

much

at the

A small

frequency.

is an average

During

are necessary

passing

energy

whereas

limited

feature

be included near

appears

no subharmonic

runs

6.

of the one-dimensional

conditions)

however,

of the peak.

probably show

and

were,

experimental

plification

peak

total

spectrum

of the propellers.

slightly

of the

intensity

the

of the initial

simulations

Another

behavior

X,Z plane.

frequency

constraints;

the

investigate

the construction the rotation

plotted

blade

although

to the

the

in figure

for future

weak,

compared

in the

to the

directions

6 is relatively small

should

a given

and

61

is given

corresponding

is not

be pointed

Two views

k2 direction,

_._ Results

vorticity

closure:

studies

by the closure. would

be

highly

in tile experiment

The valu-

is hard

to

_.3 Acknowledgements The authors (at

NASA

during want

would

Ames,

the

like to thank

at

CTR

summer

to thank

and

Finally,

we would

illuminating

all the local

at Stanford

the large

P. Moin,

of tiffs meeting. Tim P. Spalart for helpful

and

amount

of work

W. C. Reynolds,

authors would assistance. like

to thank

discussions

during

participants

to the

University) and

for

they

for the

also like to thank

tim course

R.

Summer

Program

considerable

did for us.

J. Kim

J. C. R. Hunt,

their

In particular,

perfect

P. Orlandi, Kraichnan

help

R. Rogallo, and

we

organization

C. Leith

and for

of this work.

REFERENCES KANEDA

Y.

KRAICHNAN LEE,

LEE,

1981,

J. Fluid

R.H.

1987

Mech.,

Phys.

107,

Fluids,

131. 30,

1583.

C.H., SQUIRES, K., BERTOGLIO J.P. & FERZIGER, at the 6th Turb. Shear Flow Symp., Toulouse. M.J.

MICHARD

& REYNOLDS, ET AL. 1986

W.C,.

European

1985 Turb.

Stanford Conf.,

University Euromech,

J.H.

1987

Report Lyon,

to be presented

TF-_4. July

1986.

62

J.P.

POt1QVET

A.,

Bertoglio

LESIEUR

M.,

, K.

ANDRE

Squires,

J.C.

S.A.

1970

J. Fluid

(,

Mech.,

J.H.

AND BASDEVANT

72,305. ORSZAG

and

41,363.

Ferziger C.

1975

J. Fluid

Mech.,

63

Center for Turbulence Research Proceedings of the Summer Program 1987

N88-23092 for of EDQNM Models supergrid

Validation subgrid and By

1.

Orlandi

1

Introduction The advantage

they

retain

on

the

among

higher

interaction

eddies

order

quasi-normal

where

fourth-order

approximation to negative

increase

introduction

of the

of an

the

T(k)= the

transformation EDQNM

_1f01

integration is given

(Crocco

B(k,p,

results

expression

of the

from

damping adopted

energy

dfl /l+a .]1-_

dT[kS_(k,flk,Tk)+(

is to be

performed

direct-

It requires

This closure third-order

of second third

order

order

damping

cumulants terms.

cumulant term

a very is based

and

eliminates

The leads that

1985) transfer

term

)3_(k,k/fl,

only

is

Tk/fl)]

in a triangle

in the

rE(p) p-2

]

q) is a geometrical

factor

the Markovianization

function

and

D(k,p,

of a certain

D(k,p,q)

=

r/(k)

completes

E(k) k2

(1)

(fl,7)

plane.

q) is the

(2)

relaxation

frequency

e -['l( k)+n(p)+rt(q)lt

rl(k ) + rl(p ) + rl(q ) the

EDQNM

closure

is

= ,,k + of Rome

E(q) q2

integral:

1

1 University

for the wave numbers

by

1-

The

(1970).

for the

& Orlandi,

E(k'p'q)=kZpqB(k'p'q)D(k'p'q)[

generally

by the

Turbulence

a particular

where

eddy

hypotheses

effect.

is introduced,

that

derived

integral.

products

closure

is that transfer

one is the eddy-damped

of Orszag triadic

models

the energy

require

models

equation

by

closure

the simplest

theory

in the

an unrealistic

models

different

the so-called

are replaced

one-point

characteristics,

These

the

EDQNM,

to solve

The

over

of Kraichnan

approximation

gives

Isotropic

_(k,p,q)

sizes.

Among DIA,

terms

energies.

unphysical

where

terms.

scheme

on the

models turbulence

of different

Markovianized,

numerical

When

closure

important

approximation,

quasi-normal simple

of two-point

one of the most

mechanism

2.

P.

p E(p)dp)

and

the

expression

64

P.

.0125

'_"1

'

' ' ''"'1

'

'

Orlandi

' '''"1

I

o°

I

I

I

II

I

/

,P

BE]

-.0125

I

oo%

El T

!

r:t

13

-.0250

(a)

IB *

trill

-.0375

,

*

t

i,¢,|

t

1

i

i

i

i

Illll

10

101

k

FIGURE aDS.

1.

lution

in time

and has

In the

energy

been

closure

Only

would q) within and the

of the

by

of only

it was difficult T(k)

summer

I

different

comparison

of the

has been

O(N

2) operations

N = 323 nodes. a continuous

is determined,

the

the

spectrum

of the

distribution

the

evaluation was

a single

evolution

of

thus

_ distribution from

is

complex

check

calculation

resulting

distribution

energy

the

better

but

EDQNM

distribution

interaction

by Wray, and

The

by

correctly

detailed

done

at

comparison

given

An even

only

, dissipation

transfer

describes

scales.

This

to infer

energy

closure

,

to the evo-

a detailed

transfer

lim-

was very realization.

is obtained

by

solving OE(k) + 2vk2E(k)= Starting

from

have

been

and

indicates

The ages then and

good

of the

numbers, transfer

high

figure

spectrum

with

wave numbers.

the

agreement

the The

in the the

energy

(3)

together term

with

at two

(1)

different

and

(2)

times

DS results. accurate

or supergrid)

range both

computed

DS,

(3)

transfer

to produce

(subgrid

an inertial

T(k)

energy

with

closure

model

by introducing between

used

1 shows

EDQNM

as a closure

be obtained

for the

initial

very

ability

its use

Reynolds

the

solved,

extending

a subgrid

and

transfer

in a DS.

spectra

encour-

A simulation

for at least

one

at high

decade,

a supergrid

model

scales

and the

unresolved

scales

spectrum

is then

subdivided

into

i

il

l

100

possible

energy

program the

of the

EDQNM

are

were limited

as turbulent

ill a DS and

among

plane.

such

CTR

DS,

comparisons

prediction

the

a DS requires

transfer

I

[] EDQNM

(b) t = 4.34,

simulation,

these

quantities

accurate

transfer

direct

past

scales

that

(fl,7)

to a simulation

Once

the

be obtained

in the

sparse

part

between

of energy

_(k,p,q) ited

first

(a) t = 0.54,

and

In the averaged

to demonstrate

mechanism

E(k,p,

EDQNM

transfer

made.

sufficient

distribution:

numbers.

of globally

skewness.

of the

transfer

between

low Reynolds

I

k

Energy

Comparisons very

I

10

might

to account both three

at low regions

EDQNM

for

subgrid

and

log E

supergrid

Models

65

q

SUPER

l

khi

I_

I SUB

k

l-

I

kl o

I

T

l

I

I

klo

khi

2

FIGURE scales.

2. (b)

as figure

(a)

Partition

Partition

of the

2a

shows.

decomposed

into

In

log k

of the

Tc(k)

the

DS

T, gs(klkto,

khi)

(supergrid

T(klkhi)

number each

inside

of wave is set

quadrilaterals

of points

point

is calculated,

into

per the

to 0 unless

octave

used

domain

numbers

_,(k,p,

of

p and

in the q. When

p or q is less

than

q) is set to 0 unless

p or

khi.

and

EDQNM

unresolved

q).

involving

to interactions

represents

way

Figures

T>(klkc)

and

khi),

between

involving

spectrum.

To compare the transfers tegral with those obtained been

resolved

calculated.

(supergrid)

In a similar

q is greater

interactions

is calculated

to represent

Tog,(klkto,

due to interactions

and

close

sizes

into

< khi

P

khi

for _(k,p,

k

due to interactions

transfer

be easily

integral

k_o
(klkhi)).

energy

domain

T(k) where

w

klo

due to unresolved by DS, the flowfield

the transfers 3 and from

small

DS are and

across

4 show

that

in very

large

a DS can be used

times.

scales obtained of a DS with

several the good

cutoff

distributions agreement

As a result

to simulate

decaying

wave

by the EDQNM a 128 s resolution numbers

k_, have

of T(k), with of this

turbulence

been

T(k]kc),

at t = 0.54 ....

T(klk_),

A T(klkc),

t = 4.34 T(klkc),

A T denotes

an ensemble

average.

-< ] >, The

(2)

fine-grained

pdf

is conserved

o,1+ 0 (O,v,])+ 0 (0,_,,_]) = o, where

of

at (x, _),

/(v,w;x,t) where

vorticity

be a

(Lundgren

] - _(v(x, t)- v)_(w(x, t)- w) denotes

was

flow.

in a turbulent

and

low Reynolds

Oi = O/Oxi

and

an incompressible transport equation:

Ot = O/Ot.

Newtonian

1 Dept.

of Mech.

Engng.,

University

2 Dept.

of Mech.

Engng.,

Stanford

Averaging

fluid,

in the

of California, University

of (3) absence

Davis

and

the

of body

(3) use of the forces,

lead

balances to the

for pdf

148

Mortazavi,

Kollmann

and

Squires

0 Otf -4- ViOif

0 2

= _i(
f)

-4- vV2 f

02

OV_OVj(V < Okv_Okvjl... 02

2--

OEOwj(V

< O_.v_Oku,jl... > f) 02

Ol_5OW (V < Okw_Oku,jl ... > f) 02

O_jow

(Y_W_f)

OW_OV (< _jv_Sjkl...

0_

wjvio_w_l... > f)0 --'ijk-_ii(
f)

0

ow_ow(
f)

condition

O_fkl...

(v = V,w

equation contains flux of f due to the

_-_(< f_l... > f) > f),

= W)

(4)

and

Sit

three dynamically fluctuations of the

=- ½(Oivj

different pressure

+ Ojvi)

groups gradient

denotes of terms.

1

F_-< "-0_vlv= V,w = W >

(5)

P acts

on

contain

f

in velocity

space

the

pressure

in explicit

only,

since

form.

the

This

vorticity implies

lim Fif IVl--*o_ no matter

what

will focus

value

our attention

The conditional ing and ables

W

viscous

(V,

W).

stresses Hence,

we consider

in order

to begin

of equation

caused

is considered.

are functions they

of the

are functions

In this

a manageable

vorticity

space

number leads

pressure

point

(x,/)

of up to ten

expectations

f Fi(V,W)f(V,W)dW and

does

not

= 0

by the fluctuating

conditional

with

(5) over

equation

preliminary

report,

we

on Fi.

fluxes

quently,

for vorticity

transport that

gradient, and

the conditioning

independent

number

of independent

variables.

f Fi(V,W)f(WlV)dW

thus

Fv(V)fv(v)

= f Fi(V,W)f(V,W)dW,

where

F_(V;x,t)

variables.

with increasing

to

= fV(v)

vortex

= f FdV, W;x,t)f(WlV;x,t)dW

stretchvariConse-

of conditions Integration

Single-point

pdf

of velocity

and

denotes the conditional flux in velocity space the conditional pdf of vorticity given velocity.

vorticity

149

irrespective Then,

=
.

(6)

P Integration

over

conditioned

with

parts

of the

a single

velocity

space

leads

furthermore

to expectations

variable,

F/(Vj;:,,t) =< a-O, plvj =

>.

(7)

P These

quantities

stitute,

are

therefore,

The

direct

Moin

(1987)

data

base

for the

steps:

vorticity one

the

high

values

and

so the

collected The

and

one

near

vorticity

the

used

range center

from

no conclusion statistical The

their

sample,

and

pdfs

show

correlated

that

pdfs

probabilities

that

varies

> are

shown

over

the

8 shows that

values

joint no

V2; the

with

shown form the

1-3.

in Figures

7. Note

of adequate

sample,

region

of pressure-gradient correlation

fluctuation

pressure

components

The

between gradient,

that

the

two.

in the

single on

strongly

value

(ex-

expectations compo-

in modeling.

on both sample,

of change vorticity the

The

conditioned

velocity

of importance

Consequently,

conditioned

the

conditional

linear

compo-

more

a mean

of adequate

and

inadequate

by symmetry.

show

behavior found in Figure 7, and that the rate component is independent of the velocities.

pdfs

hence

of the iso-probability

of 01p conditioned

in the

and

velocity

V1 and

of Vi, with

are

and

are not exactly with

one

V3 is zero

they

fluctuation abscissa

shaded.

with

They

are

diagrams.

point,

Areas

shapes

an observation

expectation

straight

are

velocity.

that

to the The

01p and

values

lines,

Samples

zeros

with

4-6.

of the contour

one data

there.

skewed

at different

conditioning

of the

so their

is only

with

of velocity

positively

correlation

in Figure

are

The

one

spaced,

look

values

values,

with

omitted.

maximum.

component

correlated

are

jagged

behavior

uncertain

in Figures

are

contours

discernable

of the

highly

shape

to the

there

on the statistical

same

contours

these

with the linear to each velocity show

range

extreme

in

components,

equally

the full range

to maximum

At each

01p is weakly

of Opl

< 01p]vi

The

over

with

rather

was done

velocity

are

ranges

minimum

pressure-gradient

are of the

pectation)

Note

the curves

hence

of the

negatively

conditional

Figure

plots

V1,1/'2, or 1/3 are shown

contours

nents

the

This

was the

components).

lines is the

coordinate

produces

can be drawn

joint

aspect

the

con-

by Rogers

(2) conditioning (two

or two vorticity

important

out

C128U12)

fluxes.

component;

The

mininmm

figure.

conditional

contour

from

(case

two variables

which

ranging

of the

with

and

and they

flow carried

(1986)

iso-probability levels

bins,

package

and the

component

center.

shear

Reynolds

with one velocity

follow,

encountered,

ordinate

pdf's

evaluation

investigation. linear

and

(3) conditioning

that

to numerical

for the

Moin

of the

flow-dependent

plotting

nent

evaluation

in discrete

values in the

Rogers,

(1) conditioning

plots

accessible

point

of a homogeneous

and

component;

velocity

In

easily

starting

simulation

and three

most

the

1/'1 and

V2.

consistent with

respect

components

the

expectation

local

fluctuation

Mortazavi,

150

Kollmann

I

I

I

I

'

I

!

!

I

and i

Squires I

I

!

I

I

I.,,=

X co eL

!

I

v1

FIGURE

1.

Iso-probability

lines

•

I

I

of f(vl,01p)

I

I

in a linear

i

shear

i

i

i

I

I

I

I

I

I

I

.

i

!

X co eL

V2 FIGURE

2.

Iso-probability

lines

of .f(v2,01p).

I

I

I

!

i

!

X ts co

!

v3 FIGURE

3.

Iso-probability

lines

of/(v3,alp).

flow.

Single.point

pd.f of velocity

and

vorticity

151

X el

vl FIGURE

4.

Iso-probability

lines

of the conditional

pdf

f(a]plv]).

pdf

f(O]p[t'2).

X _L

.

iiii_iiii

v2 FIGURE

5.

Iso-probability

lines

of the conditional

w'-

X et

v3 FIGURE

6.

Iso-probability

lines

of the conditional

pdf/(O_plvs

).

152

Mortazavi,

Kollmann

.4

and

i

Squires

i

i

I

I

i

.2 O.

c_ iii

z 0 m Ic_ z 0 (J

0

-.2

--o4

t

-1.0

-.5

2;--

7.

Rescaled

x a=l;

< O]pJv_ >.

expectations

....

a=

--a=3.

vorticity,

are

essentially

expectation

of the

component

V2 and

tion tions

is at zero, the

are

of the

of velocity pressure

and

vorticity.

gradient,

conditioned

with

the

velocity expecta-

W2.

portion

streamwise The

in the

variamiddle

vorticity. shear

gradient,

component,

In addition,

flat

of the

the

of the

I47] and

the

of homogeneous

pressure

velocity

that

expectations

and

contours on

1411. Note

the

sample,

the

conditioned

components

on the

fluctuating

in the

O]p

10 shows

study

9 shows

component

vorticity

to inadequate

preliminary

is linear

gradient

Figure on the

due

Figure

vorticity

no dependence

this

expectation

example,

pressure

streamwise

conditioned

indicating

component, dent

streamwise

end

In summary,

For

of vorticity.

gradient, at either

zero.

the

is independent

pressure

the

conditional

1.0

.5

VEL/VELma FIGURE

I

0

conditioned

and

the

that

work

a vorticity

flow has

the

shows

with

coefficient that

component,

the

shown

that

a velocity is indepen-

expectation

is essentially

of zero.

REFERENCES LUNDGREN, ROGERS, neous ROGERS, eling

T.S.

1967

M. M.

& MOIN,

turbulent M.

M.,

of the

Phys. P.

flows. MOIN, Mech.

1987

J. Fluid P.

hydrodynamic

shear flow. Dept. California.

Fluids,

10, 969. The structure Mech.

1'/'6,

&: REYNOLDS, and Eng.

passive

Report

No.

W.

of the

vorticity

field

in homoge-

33-66. C.

scalar TF-_5.

1986

fields

The

structure

in homogeneous

Stanford

University,

and

mod-

turbulent Stanford,

Single-point

pd] of velocity

and

vorticity

153 ®

b v2

7 i

v1

FIGURE

8.

Conditional

expectation

< 01p[vl,v2

>

w1

v2

FIGURE

9.

Conditional

expectation

< 01pJv2,wl

>

ORIGINAL OE EOOR

PAGE

IS

QUALITY.

154

Mortazavi,

A

Kollmann

and

Squires

A

¢-

¢q

old

i'D

v

v

w 1, w 2 FIGURE

10.

Conditional

expectations

< Ozplw_

>. --

a = 1; ....

a = 2.

Center

for

Turbulence

Proceedings

of the

155

Research

Summer

Program

1987

Overview Reynolds

It is well able

recognized

future

that

to simple

model,

direct the

the

experimental

models

was

possible.

to be modeled process

of the The

invited

and

Prof.

J. G. Brasseur J. Y. Chen

(Imperial (Sandia

H. Ha Minh

Prof.

J. C. Hunt

Prof.

C. G. Speziale

Prof.

D. Vandromme

M. J. Lee (NASA S. Lele

Dr.

N. N. Mansour

Mr.

U. Piomelli

Dr.

M. W.

NASA

(NASA

objectives

models.

were:

Langley) Laboratory)

Ames) (NASA

Ames)

(Stanford

Rubesin

University)

(NASA

Ames)

K. Shariff

(NASA

Ames)

(Center

for Turbulence

Dr.

J. R. Viegas

(NASA

most While and

modelers

work

was devoted

of this work Kim

Rodi

Research)

Ames)

of the some

Mansour,

Reece

closure

be achieved.

The

Ames)

T.-H

other

test

that

were:

Dr.

various

group

and

can

models.

terms

Laboratories)

Mr.

Launder,

formulas

closure

way of testing

the

of

of the

(Technion)

Dr.

(e.g.

with

No

because

assessment

to compute

a

validity.

pressure-strain, no direct

can be used

(NOAA/ERL/Aeronomy

Dr.

at Ames

its

University)

National

participants

models.

on formulating

(CORIA-CNRS)

local

As expected,

relied

equafluid

College)

(ICASE;

The

lence

of engineering

(Cambridge)

M. Wolfshtein

Shih

forsee-

(IMF-CNRS)

J • Weinstock

Prof.

data

modeling

(Clemson

Prof.

Dr.

for the

flows

to assess

for the

were to develop

in the

had

therefore

systematic

Group

participants

and

comparison

to a more

Modeling

P. Bradshaw

terms,

simulation

direct

lead

Prof. Dr.

to compute

in a simulation

are available

these

Direct

should

Turbulence

the Group

will be limited

development

is used

measurements

This

In order

model

model

in measuring

need

simulations

flows.

Traditional

then

difficulty

turbulence

by

models will have to be used in formulating the governing are a pacing item in the development of a computational

capability•

closure

full

fundamental

interest, turbulence tions. These models dynamics

of Research Stress Modeling

and

to the assessment

already

been

Moin,

1987,

to appear

the

summer

school

(1975),

to test

had

their

models

against

carried

of existing

in J.F.M.) provided

full turbulence

turbu-

out by investigators for the

model

an opportunity simulation

of for

data.

156 Weinstock the

slow

duced

& Shariff

term.

by the

energy;

They

found

theory.

component

evaluated

features

Wolfshtein

that

theory

many

significantly

features

that

the "Rotta"

with

are well predicted

& Lele

studied

of Weinstock

of the

The data indicates

and changes these

the

strong

of the

1982,

slow

1985)

term

coefficient

temporal

by the

the structure

(1981,

are

varies

variations

for

repro-

between

of the

kinetic

theory.

of two-point

correlations

(velocity-velo-

city, velocity-scalar, and scalar-scalar) with view towards improving turbulence models. They tested the linear two-point correlation model of Naot, Shavit & Wolfshtein not

(1973)

only

and

found

to the

Reynolds

Mansour

& Chen

that

it is necessary

stresses,

but

also

to relate

to all

the two-point

correlations

velocity

mean

mean

and

scalar

gradients. Shih,

pressure-strain

terms

evaluated

(Shih

and

to be published). In general, for the cases of homogeneous models

were marginal

In addition used and

stress

Reynolds

stress that

model

the

(1987). models

cases, while second-order were not tested for the In addition Piomelli the

testing,

certain

the

rapid

that

close

correlation

to the

length

A detail Lee where

study the

wall

of the

the

well for the

were

term

devoted

where

dition, events

instantaneous were found

concentrated In order Mansour bulence

shear

perform

poorly

for these

shear

case

&=

In particular,

velocity

was

but

Mansour,

term.

mean

equation

a Reynolds It was found

by Bradshaw, strain

Moin

gradient

tested

using

the

wall this approximation works well, for the failure is attributed to the varies

rapidly

as compared

to the

gradients. flowfield

transfer

rapid and justification

& Chen

was

carried

out

by

Brasseur,

by pressure-strain-rate

was

&

investi-

slow parts of the turbulent pressure for current modeling procedures.

were In ad-

events of high transfer regions were studied in details. These to be highly localized in space and are imbedded in regions of

vorticity. to gain insight & Rogallo

in a rapidly

is to shut

velocity

numbers.

the local

Poisson

gradient

energy

gated. It was found that the uncorrelated; providing strong

Mansour

strain.

case of Rogers,

homogeneous

to the pressure

velocity

of a homogeneous

intercomponent

Shih,

tensor

1987,

plain

Ha Minh coded

of the

fluctuating

under

and

anisotropy

rapid

& Moin,

Reynolds

channel data. It was found that away from the but close to the wall it will fail. The reason fact

turbulence

comparison,

the

term performed well contraction, but all

Vandromme,

in the

solution

Mansour

at different

efforts

to the

Shih,

and

shear

approximations

integral

and

return

the homogeneous

models perform channel cases.

approximation outside

flows

are linear

model

to testing

usual

is used

to

Viegas,

channel

that

1985,

for the

of the pressure strain under axisymmetric

by direct

to predict

Rubesin,

models

of homogeneous

models

models

to compute

existing

Lumley,

the models turbulence

for the cases

to testing

Reynolds

non-linear

off the

into the effects

carried

a direct

rotating

frame.

energy

transfer

of rotation

numerical

It was found

so that

the

on the

simulation that

turbulence

the

dissipation

rate,

of decaying primary dissipation

Speziale,

isotropic

effect

tur-

of rotation

is substantially

Overview reduced. while ory

the

It was

found

energy

spectrum

analysis

reveals

Reynolds

the

stress

anisotropy

underwent the

rate

are made

towards

of the flow

including

group

remains

viscous

of change of the

modeling

tensor

a pure

reorganization

Suggestions

pation

that

that

no Taylor-Proudman served.

of the

157

essentially

decay.

Rapid

vorticity

unchanged

distortion

field

is O(f_)

to a two-dimensional

state

the effects

of rotation

theso that was

ob-

on the

dissi-

(1987)

was

rate.

Finally, tested shear

a recent

mixing

length

formula

proposed

by

by Hunt, Spalart & Mansour, for a wide range flows. It was found that the formula works well

neighborhood

of regions

where

dU/dy

Hunt

et al.

of turbulent wall-bounded for all y+, but fails in the

= O.

REFERENCES HUNT,

J.

C. R.,

STRECTH,

stratified

flows

on stably

stratit]ed

LAUNDER, NAOT,

B.

D.,

and

D. D.,

their

use

ttow

E.,

REECE,

SHAVIT,

A.

and G.

AND

BRITTER,

in turbulence dense

J.,

R. E.

gas

models,

dispersion,

& RODI,

W.

WOLFSHTEIN,

1986

M.

Length

Proc.

scales

I.M.A.

in stably

Conference

Chester,

April

9-10.

1975

.J. Fluid

Mech.,

68,

1973

Phys.

o.[ Fluids.

537. 16,

738-

743. SHIn,

T.-H.,

and

Aero.

& LUMLEY, Engrg.,

J.L.

Cornell

1985

.

Rep.

FDA-85-3,

University.

WEINSTOCK,

J.

1981

. J. Fluid

Mech.

105,369-396.

WEINSTOCK,

J.

1982

. J. Fluid

Mech.

116_

1-29.

WEll,

J.

1985

. J.

Mech.

154,

429-447.

STOCK,

Fluid

Sibley

School

of Mech.

Center for Proceedings

Turbulence Research of the Summer Program

On Local Pressure-Strain By

The

P.

for

gradient

OU/Oy

and

the

sets

calculated, turbulence local

the

success the

gradient

variables

integral

equal

rather

than

2 and

U.

approximation

that

uses

solution

to its

value

of the

at the

Poisson

point

This approximation pressure-strain terms

of the

rapid

pressure

and

the

local

assumption.

that

the

assumption to the

its

proper

Laplacian

spatial

of the

will be valid

when

correlation

length

the

of the

where

the

to the

Poisson

velocity

fluctuating

mean

is being

in most existing to be functions of simulation

data

and velocity -0.5 between

this

is in favor

equation

gradient

velocity

velocity

for pressure,

of pressure of about

Ov'/Oz);

3

pressure

Direct

to

mean

the

equation

integrals.

solution

PIOMELLI

is implicit are assumed

(proportional

Analysis

the Models

MANSOUR

were used to evaluate spatial correlations results show that a correlation coefficient

the

compared

N.

of the

has been explored. models, where the

for the channel gradients. The

1.

1, N.

outside

N 88 - 23 1 0

Approximations of Term in Turbulence

BRADSHAW

reason

159 1987

of

indicates

varies

slowly

as

gradients.

Introduction The

pressure-strain

tions,

which

of the fluctuating elers the

"redistribution"

are mean rate

for two reasons: largest

little

with

pressure

any

are

stantaneous

velocity

solutions of the approximations)

1

Imperial

2

NASA

3

Stanford

such

as those

an approximation

in modeling

solid

probe

than

in the

attention

terms

in the

those

entire

the

the

inserted

we will discuss transport

Research

in the

next

spurious

Using

the

in-

(numerical

term

We shall

which

is commonly

equations.

BLANK

be

induce

section.

University

PAGE

cannot

these

simulations

Center

?RECEDING

and flow.

College Ames

layer

equations without any modeling of measurable and of "unmeasur-

to the pressure-strain

Reynolds-stress

models.

fluctuations

flow,

are

equations;

the

a boundary

turbulence

mod-

equations

all-important

undisturbed

equa-

components

turbulence

stress

velocity

in the

in the

domain,

from

assess

within

because

various

shear

in those

to properly

fluctuations

of accuracy,

larger

field

is available

transport

and

much

to be modelled

pressure

any

received

Reynolds-stress

fluctuation

full time-dependent Navier-Stokes can be used to deduce statistics

quantities paper

on

usually

have

data

setup,

in the

pressure-strain need

assurance

fluctuations

fluctuations

in this

that

experimental

In an experimental measured

the

terms

of the pressure

of strain, first,

of the terms

secondly,

able"

products

NOT

FILMED

discuss used

160

P. Bradshaw,

2.

Local For

an

velocity

N. N. Mansour

approximation

to

the

pressure

incompressible

flow

the

fluctuating

field

by solving

the

02P

Poisson

-

with

the

boundary

_

condition

U. Piomelli

pressure

can

be

obtained

from

OX

at the

i

I Ou_ Ou_

Ou_ Ou_ _

_ OZj

OXj

OX

i

OZ

i

(la)

]

walls

Op = l___O_v ' Oy Re Oy 2"

Equation

(la)

shows

quantities will have and a part depending and

"slow"

change the

parts

in the

split

to zero which

and

field.

However,

of each

pressure

correlations

of the

in this

for

case

pressure

the

former

with

will

inhomogeneous the

case

on the

other

fluctuating

can be split

boundary

of the right

respond

fully

hand

in three

condition

developed

side

parts:

of (la) the

at once

rapid

to a makes

channel

the

will integrate pressure

pl,

satisfies 02p I OxjOxj

with

of the

only

An

terms

(lb)

to the mean velocity gradients OUi/Oxj fluctuations. These are called the "rapid"

because

velocity

obvious.

integrals

any

proportional on velocity

respectively,

mean

less

volume

that

a part only

the

equation

2 0Ui Ou_

O_jO_,j

and

the

boundary

condition

at the

_

2 0Ui Ou_ Oxj Ozi

(2a)

walls pl

-0;

(2b)

0y the

slow pressure

p2, which

is the

solution

to:

(3a)

] with

the

boundary

condition

at the

walls

aPl

oy

and,

finally,

the

boundary The

the Stokes

pressure-strain

in p, so that

pressure

condition

the total

(lb) terms

term

pS, which at the

that

O;

(3b)

is the solution

to Laplace's

equation

with

walls.

appear

in the

Reynolds

stress

equations

are

will be equal to the sum of the rapid pressure-strain,

linear slow

On Local

Approzimations

I?

of the

Pressure.Strain

term

161

o. /;......./-_.......................

o

50

lOO

15o

y+

FIGURE

Pressure-strain

1.

pressure-strain condition (2b)

term

and Stokes at the walls

in the -_-_

pressure-strain. is

p] _ in which the

G is Green's

walls.

added

Note

function

that

most

with

Neumann

homogeneous

boundary

is consistent

with

the

(4),

the

the

entire

on

rapid

the

local

gradients

pressure

domain,

is an integral

mean

velocity

the

of the

gradient,

of the integral

which

walls

by the

allows the

1 _ OUi

Ou_

conditions

terms

that

for the

modelers.

mean

is often

in (4) to yield

boundary

one

that

to take

following

rapid

be The

pressure

Although,

velocity

made

at

should

are used for the pressure.

weighted

assumption

boundary

integral

at the

used

the

(4)

Neumann

(2b)

with

Gdl;

conditions

condition

of (2a)

surface

approximation

in modeling

outside

the

equation.

solution

homogeneous neglect

to (4) if non-homogeneous (4) and

The

1 Iv 2 OUi 4rr 0zj Ou_ Oxi

modelers

use of the

transport

by

gradients

over

pl depends

only

the

mean

approximation

velocity for the

pressure: pB

Direct

experimental

of direct

simulation

computing p].

The

the integrals purpose

for channel 3.

measurement results

in (4) and

of the

flow

of either

allows

present

us to (5) and

paper

at low Reynolds

'd

p] or pB

test

the

is impossible,

approximation

comparing

is to examine

the approximate the

results

but (5)

value

of this

the

use

directly

by

pB with

comparison

number.

Results The

results

of the

direct

Re,. = 180 (Kim, Moin presented in this work.

and

numerical Moser,

simulation 1987)

were

of fully used

developed

to compute

channel all the

flow at statistics

162

P. Bradshaw,

N. N. Mansour

and

U. Piomelli

X I! II

t'v

FIGURE

2.

Figure

point

1 shows

transport rate

Two

the

equation.

positive

The

Note

accuracy

prisingly within

(y+


) will

averaged

of coherent

between as large

by

in a so-called

is strong

regions

(at

appropriate

to the wall,

resulting

deviation twice

(denoted

time-step

the

simulation

conditionally close

low- and

distance

amplitude

which

wall,

results

of the

and

averages

back

structure

half-channel

variance

An ensemble

individual

entire

(the

of maxinmm

7a.

at one

of the and

by u,- (or other

feature

channel

averaging)

experimental

conditional

normalized

interesting

of the centering

180 high

point,

between

sides

conditional

and

the

are used

involves

z (as in conventional

600 long,

levels

sublayer.

region

layer.

is the in Fig.

Similarly,

on the

downstream

The

uv-contribution

side, on

242

A.

V. Johansnon,

P. H. Alfredsson,

and J. Kira

10 2

d.d'O

n+o"4-

+ ÷D n 10-3

10 -4 •

w

|

|

•

u

•

•

10 0

|

|

•

|

i

•

g

u

101

g

10 -2

t

FIGURE

5.

probe. the

Number

Simulation

same

It was found

merical,

presented

fact

In the

that

the

develop

reduced

values

of the

layer.

The

scale

at

to find to retain the

shown

an

the

resulting

in Figs.

at about

uv-peak

and

is rather

procedures,

of the

of the

structure

spanwise 8 was

mean

than

the

asymmetry,

often

an asymmetry

to the sign of the illustrate

found

local

without the

spanwise

found was

gradient

a mechanism

in the

imposed at the

for creation

noted

that

as well as nu-

spanwise

detector

centering.

resulting

jitter

position,

or without

and

the

spanwise

cen-

of the flow in the spanwise other hand, often tend to

velocity

symmetric

resulting

small.

gradient to be

the be

to the

with

gradient.

and

It should

downstream.

streamwise

reasonable as is illus-

experimental

obtained

propagate

(absolute) 7 and

or 2/3

average,

averages

(+)

to obtain

applied,

can be attributed

uv-peak

as they

gradient

spanwise

side.

results,

are the result of homogeneity shear-layer structures, on the

of the

asymmetric

conditional

according

wall,

the

averaging

asymmetries spanwise

the

by a stationary

contributions,

ejection

been

centering

of the

is essential stress was

on the

between

conditional

average

events

dient

time

oil channel

centering

all previous have

no spanwise

spanwise

strong

G6ttingen

Reynolds

no such

knowledge)

tering, symmetric patterns direction. Instantaneous

the

latter,

separation

In conventional

per unit

of the events

associated

in the literature

with

spanwise

be detected in the

centering

of the

authors'

reduction

in the

spanwise

9a,b.

present

would

measurements

uv is drastically

the

that

number.

that

by Figs.

The

and

estimates

maximum (to

(o)

Reynolds

quantitative trated

of events

This at the

at the about

center

0.20

of the

shear

point mean

8a.

events point

the

likely

In order

and

also

in

z-coordinate

(Fig.

streamwise

for gra-

it is more

in Fig.

individual

of high

large

of the

by switching detection

give

detection

In reality, pattern

can

10).

The

gradients

Shear-Layer

Structure8

243

CENTER OF SHEAR-LAYER

-200

FIGURE in

6.

channel

detected

Conditionally flow

streamwise

at

with

(
/u,.ms(V))

200

in

the

near

& Eckelmann, in the

wall

1987)

symmetry

region showing

plane.

Events

k=l.

150

100 y 5O

0

(a)

150

100 Y (b)

:::::22222222 ...............

150 I_ I

//"

"'_

100

50 -300

-200

-160

6

160

200

300

X

FIGURE drawn

7. lines

Ensemble and

(a)

< u >/u,.,n,(y),

(c)

< v >,

contour

averaged

broken

lines

contour increment

VISA-events denote

increment is 0.1.

positive is 0.5.

detected

with

and

negative

(b)

< u >,

L=200

and

contours, contour

k=l.

Full

respectively.

increment

is 0.5.

244

A.

V. Johansson,

P. H. Alfred_on,

and

J. Kim

-100 ....--

j

-50-

.= -

s

"'-- 2".'_

""

i:::i)iiiiil;i 50

[ia_.....::L...:.....................

100

-100

.,'

-50

,- .......

:--,- ...............

,

"-. ".

0-

_;"

2_X2:

2 2

"- .....

,........ ::::..... ,-"''--A:H-__:... __+__:%%:'_ ....

,

......... :

,'.......

-1 O0 •

-',

-50

,-""

..................

:'

....

.

-.°.

....

.....

.--

"...............

-.

.....

+ _..

.....

50

- .......

- ,'

,-+"

..--:"

--.

......

..

.

""---...

.-'...

i:'

..................'

_ '

--....

..----..

----'"

...................:.:::.............

--"

'..

" +

j ........

+ ,

"'-;.

.-.

..; s-+-.....

(c)

":'::"

100

-3oo

_.................

',

-2+o

.: ...........

-16o

-->

. .........

o

_

1+o

...........

2oo

X

FIGURE shear

8.

7).

layer (a)

< uv

Contours structure.

< u >, contour

>, contour

in the

(z, z)-plane

at y=15

parameters:

L=200

Detection increment

increment

is 0.5.

is 0.5.

(b)

< v >,

of the and

k=l

contour

conditionally (same

data

increment

averaged as in Fig. is 0.1.

(c)

Shear-Layer

Structures

245

(a)

150 100 y i •""

50-

,'-..

/ .'-I""

"

" """"

7",

° It'

t o" ""

......

•

..............

0

I

..o°" .

...

-

,

-"

"°

_1

o.°-'"

. ................... .......

..":':-:-."::- :::: ." -"

• ._._._===---_

...................... i

150

(b)

100

J

-..°-°°'_•

-.

y

ss

"•

r

,,'.

•'---''''-.

_,

#l 0-

_

._-'''°''>"

.................................

i

I

-__.:. -_-_-.'. -"

0

-zoo

-300

-lOO

0

2()0

100

300

x

FIGURE (a) with

9. Conditionally averaged uv (L=200, k=l, contour alignment in spanwise direction (b) without alignment.

-loo I- ........ I

,

- .....

increment

is 0.5).

..... ,;.....,

,...............

z

50 ;.--;: .................... 100

:. ........

-300

-200

-100

_ ()

160

260

300

x

FIGURE

10.

Conditionally

average

(
)

(L=200 and k=l) and the asymmetry condition at y=15 and contour increment is 0.5.

through a relatively production

spanwise

motions

of the high-

detected

(same data

and low-velocity

important role for the generation (as will be discussed below).

with

regions.

of conditionally

the

VISA-criterion

as in Fig. 8a). Data

Such gradients averaged

is

play

turbulence

246

A.

The often space-time is chosen

occurring

P. H. Alfredsson,

asymmetric

features

development, of a single so that the center of the

clearly

shows

whereas

the

resulting

the sharpening planar

issue

in the

of the

streamwise

for the

Ov

--

only

term

However, higher due

the than

that total

what

to strong

streamwise Eq.

remains

gradients

velocity

(1),

which

averaged

over

be

is illustrated an area

for

x-

12c).

dissipation

and

The

14).

left-hand-side spanwise imply

3.

here

expression gradient

contributions

(1).

Hence, of the

15a,b).

viscous

units

were also

In order

and

structures

in the

is the

spanwise

is mainly averaged

second terms

extent,

vicinity

averaged although

shear

of the Ou/Oz

there

layers,

exist

they

have

in

been

respectively). over this Moreover,

shear does

layer not

high

do not

term

(Fig.

enter

values

the

of the

in themselves

production.

for a turbulent analyzed.

to investigate

One

may

for the

note

higher

inner regions are mainly

that

Reynolds

Reynolds number dependence flow on the scale of inner-layer outer

and

This

conditionally

production

dU/dy.

is substantially

12b).

terms

the

turbulence

determined

These

data

bases

flow at two differare

described

the size of the near-wall

in detail structures

variation, the ensemble-averaging procedure Reynolds numbers (Reo) of 670 and 1410 the

structure

number,

indicates structures,

of the flow.

boundary-layer

whether

shows any significant Reynolds number was carried out for momentum-thickness (Figs.

flow

>

out to y = 30 gives an average as the long- time mean value.

edges

associated

12a)

(Fig. of the

13 where

is < uv

(Fig.

term

of these

conditionally

data bases

numbers

(1987).

is their

conditionally

results

Computer-generated in Spalart

flows

the

two-dimensional

sense

y-directions

the

of Eq.

to the

Boundary-layer

ent Reynolds

this

is large

of u at the

shear, regions,

Ou

production

energy that

that

for a plane

of 300 × 40 (streamwise

It is noteworthy

mean

low-speed

(1)

largest

in Fig.

of turbulence

of the

and

show

> -_ii(
.

conditionally can

the

in turbulent

can

energy

-v_-W_


p +

2

>;
;

(m)

centered

the

of

details

(e) OvlOx

/0xSzz

structures


; (i)

at

=

125.

structures.

(f)_<
=

-

Note

(a) >;
;

2 >O/Oy;(k)=+< $22

that

>;

(j)

284

A.K.M.F.

Hussain,

J. Jeong

and

J. Kim

175

(e)

(f)

155

\, "\k

,

.--.

-_.

/..p

"\

\)

\\

" -.

135

,.-.

i.--_1

..... ,.

".,,j

t

Y

-I

+"

"

I

li

115

•

/

-

)

_

'_

k

:,

l

}

:'/

•

- .....

C, .!,b

_

."

1

/

c,

-4O

175

(g)

-2O

f

0,54

/

(

,

\

0 +

__"

( 20

40

X

/

155 /

135 y-I-

115

0

95

< / \.g

75 -40

:£ -20

"_ 0 +

2-

" 20

40

x

FIGURE

',r

/

95

75

,

"

3 .... continued.

......-,,; ,.,, PA.Gg

Ci t'_.'-, kxJ_ ;': "-_

o_" _ooR- QUA._

I_

---

ORIGINAL

PAGE

IS Structure

OF

155

POOR

j

of

Turbulent

Shear

Flows

285

QUALITY_

%

:;

,

_.

y+

175 (k)

(J)

155 _

%"

135

_

...........

-.

,-

y'_

115 "

o,_, "., ""0"__i'" '/

-1

9s 75

_" ,1 .

/ , / -40

o

,/ -20 ,

_

,

i

". 0

...

',

x+

FIGURE

3 ....

continued.

....... 20 '

.. 40 ;,

-40

-20

0 x+

20

40

286

A.K.M.F.

175

Hussain,

J. Jeon9

and

J. Kim

\

(I)

\ \ (13

155

135

..,-*:-._...-- .....

,-..-.

"

. i ")

+ Y

?, _ -

115'

_/

/

"¢,._\\'.' : ,-....../.',,_,t

_'-,,o

/

",', "..

',

",

,

'

,, dD,

" :D-,

"" ',

95 -_

// \

',

6"

/ /i /

75

.

/

-40

",

-20

3 ....

and they

20

tion peak, as follows:

4O

40

+

continued.

are cross-correlated, one can

obtain

then

from

streamwise

the

locations

(_zm,

and transverse

@m)

of cross-

propagation

correla-

velocities

Uc, Vc

_Zm Uc-

(t2 -- t,)

Vc-As a first velocity

step,

(ui),

U_ for ui and (_i).

Figs.

we have

pressure

perturbations mean

region,

but

convection velocity

how

values

same the

each

other.

being

only

higher the

velocity

than data

the

mean

z-correlation

alone.

velocity

wall region

To determine Vc in addition the current value of U_ may

these

lower

in the

somewhat Future

(1)

velocities

here

indicating

the

as functions

Moreover, slightly

and the

streamwise

Fig.

5a shows

presented

of pressure

vorticity,

Fig.

fields.

convection

with

being

of U< from

(_i)

profiles

profile,

between and

the

closely

(t2--t,)

vorticity

of y, and

5b show

agree velocity

differences

and

p as functions

4b and

It is surprising the

determined

(p),

/

r

,

0 x

FIGURE

,

/

than near those

4a shows profiles

of

the

of

profiles

of U_ for vorticity

of y+. for

velocity

profiles mean the

nature

and

closely velocity

wall.

reported

is consistently elliptic

correlations

vorticity agree

with

in the

outer

There

are

clear

in literature. higher of the

than pressure

The those

of

field.

eztensions

to Uc. When correlation in (z be somewhat different from that

and y) is used, found from the

Structure of

1.0

Turbulent

(b)

Shear

Flows

287

.............................................................. _:

Up

0

.5

1.0

y/_

20

- (a)

15

o..°'"

10

5

0 2 × 10 -1

101

100

102

y+

FIGURE

4.

Propagation

(b) in wall coordinate:

velocity --

for velocity

, u; ....

and

, v; -----

pressre , u,; ........

(a) in global , p.

coordinate,

288

A.K.M.F.

1.0 f (a)

Uc,. .5

Hussain,

J. Jeong

and

_

J. Kim

..............

"'

I

"

I

I

I

0

20

I .5 y/8

I

I

I

I

I 1.0

(b)

15

10

"

I|

0 2 × 10-1

100

101

102

y+

FIGURE coordinate:

5.

Propagation --,

wz; ....

velocity , a_;

for vorticity -----,

tVz.

(a) in global

coordinate,

(b)

in wall

Structure

of Turbulent

(2)

Determine

Uc and Vc for positive

(3)

Determine

U_ and V_ values

C.

Taylor The

hypothesis

turbulence

obtained number

and negative

for all variables

of frozen

for inferring

289

wz separately.

in the two homogeneous

and

higher-order

turbulence

spatial

by a stationary probe. spectrum from measured

dissipation

Flows

is a common

structure

assumption

of turbulence

moments.

While

there

have

hypothesis

assumes

is the

{ui,p, wi,c}. sain, 1981):

Taylor

The

+UT

advection

value

of UT used

and

in the

mean

Determine

first,

second

and

Taylor (3)

For

vorticity fields using true vorticity fields. which

for

fourth

from

choice

be

any

of the

varied

in the

(Zaman

& Hus-

velocity

velocity 1987

CTR

session.

It is seen

in the outer layer, except of DT¢ with the smoothed the

variables

hypothesis

that

very close contours

is not directly

to of

associated

eztensions

Taylor

hypothesis

of UT produces

the

and

minimum

velocity

fields,

error

in the

and use

comof the

hypothesis.

situations

in the

hypothesis

velocity

filtered

Future

(2)

of this

has

propagation

ui, wi, p, it is found that the departure with large-scale structures.

Compute pare with

discussions

velocity

local

both mean and rms values of DTdp are small the wall. Comparing instantaneous contours

(1)

extensive

known to Taylor himself, The direct numerical sim-

¢ can

literature

= ui( , y, z), used the

data

)¢=0

velocity

UT:U(y), = u(x, y, z), = We have

in experi-

temporal

that

= UT

from

been

ulation databases allow us to make a thorough evaluation different flow variables as a function of shear rate.

where

flows.

It is also commonly invoked for inferring wave frequency spectrum as well as in measurements of

of the limitation of the hypothesis, which were obviously no direct test of this hypothesis has been possible yet.

The

shear

hypothesis

Taylor

mental

Shear

when

convective

which mechanism) DT¢ = O.

there

are

balance

large

values

equations

produces

of ¢.

the maximum

of DT¢,

evaluate

Determine

the

which

contribution

term

to the

various

terms

(and

hence

departure

from

REFERENCES CANTWELL,

transport 136, 321.

B.

&

in the

COLES,

turbulent

D.

1983 near

An wake

experimental of a circular

study cylinder

of entraimnent . J. Fluid

and Mech..

29O

A.K.M.F.

HAYAKAWA, the

M. & HUSSAIN,

turbulent

plane

University, HUSSAIN,

7-9,

HUSSAIN,

1983

Notes

in Physics

of coherent

Turbulent

structures

Shear

Flows,

in

Cornell

(ed.

J. Jimenez),

Springer-

1986

structures-

Coherent

Reality

structures

and

and

Myth.

turbulence.

Phys.

Fluids.

Fluid

Mech..

J.

& ZAMAN, excitation.

K.B.M.Q. Part

1980

2.

Vortex

Coherent

pairing

structure

in a circular

dyanmics

. J.

jet Fluid

101,449.

HUSSAIN,

A.K.M.F.

nized

& ZAMAN,

motions

HUSSAIN,

in the

A.K.M.F.,

axisymmetric J.,

KLEIS,

P.

_

M.J.

_

neering,

REYNOLDS,

R.

a turbulent August

W.C.

HUSSAIN, mixing

Part.

1987

1985

M.

1980

Turbulence

Numerical

159_

98,

in fully

177,

in an

developed

133. on the

Department

spot

97.

statistics Mech..

of orga-

85.

A turbulent

experiments

TF-24,

study

Mech..

Mech..

J. Fluid

No.

structure

of Mechanical

of Engi-

1985.

A.K.M.F.

5th

J. Fluid

2.. J. Fluid

number.

Report

layer:

layer.

An experimental

SOKOLOV,

R.D.

University,

and experiments,

& MENON,

A comparison

Symposium

S. 1985

between

on Turbulent

Coherent

direct

Shear

structures

numerical

Flows,

in

simulations

Cornell

University,

7-9, 1985.

J.

&: KIM, Mech.. M.R.

neous

_

layer.

turbulence,

Stanford

METCALFE,

ROGERS,

S.J.

MOSER,

1985

mixing

flow at low Reynolds

homogeneous

Fluid

K.B.M.Q.

turbulent

free shear

MOIN,

channel

J.

1982

118,

341.

Numerical

& MOIN,

turbulent

P.

flows.

1410. TENNEKES, Press,

1987

H. & LUMLEY,

investigation

The

J. Fluid

P. 1987 Direct NASA Technical

SPALART,

of turbulent

structure

Mech..

of the

176,

1972

vorticity

flow

. J.

field in homoge-

33.

simulation of a turbulent Memorandum 89407. J.L.

channel

A First

boundary

Course

layer

in Turbulence,

up to Ro

The

--

MIT

1972.

J. 1983

Ph.D

K.B.M.Q.

coherent

the

on

Coherent

controlled

Mech..

ZAMAN,

Symposium

Lecture

A.K.M.F.

under

ZAMAN,

Eduction

303.

HUSSAIN,

Tso,

J. Kim

1985.

1980

A.K.M.F.

173,

MOIN,

5th

1985

and

1980.

HUSSAIN, A.K.M.F. 26, 2763.

LEE,

J. Jeong

A.K.M.F.

wake.

August

A.K.M.F.

Verlag,

KIM,

ltussain,

Thesis,

& HUSSAIN,

structures.

K.B.M.Q. axisymmetric

The Johns

J. Fluid

& HUSSAIN, mixing

Hopkins

A.K.M.F. Mech..

1981

Taylor

1983.

hypothesis

and large-scale

112,379.

A.K.M.F.

layer.

University,

J. Fluid

1984

Natural

Mech..

138,

large-scale 325.

structures

in

N88:23114' Center for Proceedings

291

Turbulence Research of the Summer Program

1987

Inflectional Region

Instabilities

of Bounded By

JERRY

the

Turbulent

D.

BLACKWELDER,

in

Shear

SWEARINGEN, 1 and

Wall Flows

z RON

PHILIPPE

R.

F.

SPALART

2

Objective The

primary

thrust

sponsible

for

turbulent

shear

(Swearingen events flows ble

strong

the

upon 1987),

streak velocity

have

many

of the

oscillatory

motions,

turbulence

production

sheared

and same

etc.,

Finlay,

Keller

surface.

may

region

The

& Ferziger

i.e.,

production transitional

1987),

growth Since

strong

the

this

unsta-

and

in the

of these bounded

shear,

a similar

layer

on the interface

direction

rapid

also experience

of bounded

In bounded

streamwise

re-

boundary

formed

of turbulence.

characteristics,

they

mechanisms

that

which fluid.

in the

production

wall

probable

profile

surrounding

1987,

or more

in a transitional

highly

velocity the

one in the

work

fluctuations

to the

to a breakdown

events

it seemed

and

& Blackwelder

perpendicular

leads

to identify

previous

by an inflectional

developed

direction

was

production

Based

low-speed

(Swearingen

flows

turbulence

flows.

preceded

profile

ities

research

& Blackwelder

were

between

of this

instabilturbulent

low-speed

type

regions,

of breakdown

and

mechanism.

Methodology From the turbulent-boundary-layer the instantaneous velocity, pressure, first

effort

the

was

IRIS

devoted

workstation.

direct numerical simulation of Spalart (1988), and vorticity fields were readily available. The

to examining

these

The

shear,

spanwise

fields

visually

Ou/Oz,

through

was

examined

related normal vorticity, w_. The inflectional profile associated with this orientation may be unstable in the Kelvin-Helmholtz u and

w fluctuations.

deemed the

average

tion

structure

method

modifying could

Hence,

to be important. based existing

associated on w using software

be conditionally

associated

1

U.S.C.

2

NASA

with

Ames

the

role

averaged

Center

with

these

of the

spanwise

fluctuations

w indicated

a VISA

programs

turbulence-

Research

the

Secondly,

and

within

to educe production

the

relevant events.

of

as the

with the shear layer sense and produce fluctuations

studied

turbulence CTR.

wonders

as well

velocity were

quadrant the

the

production.

technique Thus, parameters

the

was

to see whether A detec-

was

obtained

shear

i)u/Oz,

leading

by etc.,

up to and

292

J. D.

Swearingen,

R.

F. Blackwelder,

and

P.

R.

Spalart

U I

z

_11

1

1

1

I

¼

r

(

x

FIGURE

1.

b) normal

Contours vorticity

in an z -

z plane,

a)

Streamwise

velocity

fluctuations

u';

w'_

Results All of the results data

of Spalart

velocity by

field

regions

was

reported

(1988) near

the

of strong

typically

that

revealed On

also

both 0.5 with

large

that

typically

in the

beside

In many direction

and

region above

instances, with

the

a wavy

had

& Blackwelder

wavelength

of 100 the

up into

chaotic

undulating motion.

study

of this

w fluctuations.

motion motion Data

signs

(see

were Fig.

on the two

values.

Finlay

a time

was

its

undulated,

movement

and

revealed at

streaks

instantaneous surrounded

1),

IOu+/Oz+

sides.

Above

l

the

Three-dimensional

plots

largest

magnitude

and

was

streak.

region

As

of the

streak

< 80 it had

This

t_//u_.

low-speed

of the similar

low-speed

(1987)

200

followed, Closer

the

boundary-layer-simulation

+ Oz-----_

< y+

low-speed

the

observations

opposite

and

motion.

Swearingen

strong

15

the

sides

Oy+ showed

from

wall

0.2 and

+ was

were obtained The visual

shear.

between

streaks, Ou+/Oz of the modulus

here

for Ro = 670.

seen

was et al.

sequence to grow

that

y+ = 15 were

i.e.,

the used

moved

similar (1987), of the

in the

to those and

had

low-speed

in amplitude undulation to examine

and was

spanwise

observed region finally

associated

this

by

a streamwise

aspect

was break with of the

Inflectional low-speed

streaks.

regions

A quadrant

in quadrants

exceeded teristics: a preferred

2 and 3 of the

direction

15 ° in the

z -

z plane.

a cone

opening

to 1200

v/u,-.

Secondly,

aligned

with

undulatory tion.

motion;

Thirdly,

in the Reynolds

served

to occupy

the

uw

vw

of being

the

had

were

frequent

than

generally these

with either

denoted

events

were

Auxiliary

The

point, recorded.

2.

the

origin,

The

uw or the

of the

of the

data, the

distance

to -15

extent

of 600 were

turning

in their

changes

the

°

-3 events

its direc-

high-production

the uw events

were

corresponding

undulating

same

data.

ob-

uv events;

the

downstream,

were

quadrants

The

events

vw

they

they

were

broke

used

4 in the

with

occurred, that

in the

was

1 and

quadrants

corners

often

technique

in these

When

near

streaks

quadrant

namely,

the

is associated is directly

& Reid (1984).

were

v > 0,

much

less

however,

had

been

up into

they

lifted.

chaotic

As

motion.

of uw,

undulating

statistics.

The were

for the vw joint

uw

motion

uv,

Uc, was

obtained

in time of UcAt

vw

were

at

Ou/Oz the

data

points

earlier

could

symmetrical

the

u = 0 axis.

used

to

shown

in

symmetrical recorded

was

inflection

were

at 4-0.16

were

about

y+ = 15 plane

about

u2/v. to ascertain

be

detected

about Similar

the

in

w = 0

results

were

distribution.

by

comparing

by At + = 9. The to form

to being peaks

inflection

gradient

inflection

has

described

distributions

asymmetrical

at the

and

in the

These

then

an inflection

at the

of Ou/Ov

< 20.

and

shear

data

values

profiles,

where

of the

is close

symmetry,

distributions

embedded

U(z)

value

10 < y+

of Ou/Oz

spanwise

velocity

growing to the

of the shear

probability

probability

the

for the

range

probabilities

inflectional begin

Thus

and

obtained

in the

with

with

should

proportional

of inflection,

plane

but

point

A preliminary attempt was made to determine flow structure occurred as the flow was convected displaced

streak (i.e.,

same

A similar

vw events.

inflection

conditional

as expected,

locity,

were

of 10 to

-10

and

streaks

amplitudes

streaks

visually

results

consistent

joint

obtained

the

conditional

standard

axis,

of the

large

streaks

by Drazin

z-V

the

Fig.

the

at

a streamwise

quadrant-2

than

w motions;

had

production

Similar

construct

whether

the

the wall.

the

for points

in the

The

corners

v and

which

rate of growth

searched

points

from

at the

as described

were

over

direction

aligned

-4 uv events

extent

magnitude

results

disturbance

occurs.

the

followed

If the turbulence any

that away

regions

were

the low:speed

from

spatial

whose

streamwise

low-speed

and

sense)

regions

had three important characevents were occurring had

the

of the

where

to plot the intense

scales.

simultaneous

compared

those

events

the

quadrant-2

larger

with

locations

where

longer

This motion,

i.e.,

and extending

exact

average

lifted

the

plane.

quadrant-3

a much

events

to examine

Similar

locations

It was also observed process

an angle

i.e., at the corners

events i.e.,

made

the

by plotting

was developed

uw plane;

downstream

the

program

The instantaneous field regions where quadrant-2

that

forming often

technique

4 u_ were plotted. first, those spatial

293

Instabilities

the function

two

second f(z

data plane + UcAt,

where the downstream. planes

at

was shifted to + At)

largest The the

same

in the and

changes in the convection ve-

the

y+

value

z direction

but by a

difference/_(z)

294

J. D. Swearingen,

R. F. Blackwelder,

2[

and P.

R.

Spalart

(a)

0

0

A

0 II

_

0

0

--+ 1o,.

0

0 -.5

0

0

.5 du+/dz+

1.0

1.5

(b)

-.5

FIGURE

2.

b) du+ /dy +

Conditional

0

probability

.5 du+/dy+

distributions

1.0

1.5

at inflection

points:

a) du+/dz+;

Inflectional with

the

first

differences,

plane

f(z,

to) obtained.

g, was

used

to determine

velocity in the logarithmic within the accuracy of the vection

velocity

vorticities,

was

and

physics

and

for modeling

The

295

minimization

the

and outer calculation.

constant

pressure.

Instabilities of the

convection

velocity

mean Uc.

result

and

should

The

region of the flow followed the However, for all values of y+

at Uc _ 10 u_ for all flow variables;

This

square

could

be very

important

be pursued

of the

convection

mean velocity < 10, the con-

i.e., the

velocities,

for understanding

the

further.

Summary The and

results

support

responsible

There

was

low-speed

region.

The

frame

of the

Future

of the

and

sufficient The

data

in detail,

observed

up into

chaotic

The

as time

components

the undulating

wall

became of the

the up

low-speed

progressed,

In the last

portion

region.

persisted

to develop.

mechanism.

with

surrounding

condition

increased

u, velocity

an instability

in the

profile

this

for an instability

v and

with

be associated

streaks

that

which

may

velocity

indicated

motion

in accordance

low-speed

an inflectional

time

an oscillatory

instability

also

large

during

available

streaks

time

appeared

motion.

work

The and

an inflectional

sequences

of an instability.

to be breaking

the

shear time

indicating

motion,

that

disintegration

a strong

developed

indicative this

for the

often

to 60 u/u2_, streaks

the idea

work

future,

has one

been hopes

4 th quadrants The

fact

at about

based

to be able

of vw

that

the

primarily

10 uT has important

It is hoped

that

this

to apply

seemed

convection work

upon

conditional

to be the velocity

best

in the

implications can

a study

of the sampling

detection wall

for the

also be continued

instantaneous to the

function

region

data.

In

The

1 st

data.

to pursue.

was found

to be constant

wall structure,

as well as modeling.

in conjunction

with

J. Kim.

REFERENCES J.

SWEARINGEN,

D.,

AND

down of Streamwise Mech. FINLAY,

W.

H.,

Transition Univ. SPALART,

P.

in

R.

Ro = 1410". DRAZIN, Press,

P.

G.,

London.

KELLER, Curved

1988 J. Fluid

R.

BLACKWELDER,

Vortices

J.

B.,

Channel

"Direct Mech..

AND REID,

W.

F.

in the Presence

AND FERZIGER, Flow".

Simulation 187, 1984

1987

"The

of a Wall".

Report

J.

H.

TF-30,

of a Turbulent

Growth

and

To appear

1987

Break-

in J. Fluid

"Instability

Mech.

Engr.,

Boundary

and

Stanford

Layer

up

to

61. Hydrodynamic

Stability.

Cambridge

Univ.

N88-23115 Center ]or Proceedings

Active

layer ,By

The

terms

1987

model

M.

active-layer

linear

297

Turbulence Research of the Summer Program

T.

LandahP

model

are large

for

the

strongest

near-

near

wall region,

J.

Kim

for wall-bounded

only

in a thin

in the region outside the active flow driven by the active layer. a direct simulation of turbulent are the

wall-bounded 2 and

P.

turbulence

layer near

turbulence

the

R.

SpalarC

hypothesizes

wall,

and

that

hence

the

the

non-

turbulence

inner layer can be modeled as a linear fluctuating This hypothesis is tested using data obtained from channel flow. It is found that the nonlinear effects

the

wall with

these

involve

a maximum primarily

the

at around

y+

cascading

mechanism

= 20 and,

outside

leading

to

dissipation.

1.

Introduction Laboratory

activity

experiments

and

in wall-bounded

numerical

turbulence

simulation

is the

have

highest

shown

in the

that

the turbulent

immediate

neighborhood

of the wall, in the viscous and buffer layers. Therefore, nonlinear expected to be strongest in this region. A possible model for the

effects may be turbulent field

might

therefore

an active

layer

as a linear

2.

be to consider fluctuating

the

turbulence

flow driven

in the

region

active

layer

by the

outside (Fig.

inner

1).

Analysis We subdivide

field,

u,v,w,p.

making the

the

flow field

By

use

of the

v-component

in a parallel

elimination continuity

is found

of the equations

(Landahl,

mean

pressure in the

flow,

U(y)_il,

from

the

process,

the

and

momentum following

Los

is the

(space-time)

Los(v) and

q contains

all the nonlinear

for

(1)

Orr-Sommerfeld

= (O/Ot

equations, equation

1967): Los(v)=q

where

a fluctuating

+ UO/Ox)V2v

operator, -

U"Ov/Ox

- vV4v

(2)

terms,

q = W2T2 - 02Ti/OxiOx2

(3)

where

Ti = 0 i¢/0zj 1 Massachussets 2 NASA

Ames

Institute Research

of Technology Center

(4)

298

M.

T. Landahl,

J. Kim

and P. R. Spalart

Y, x 2 LINEAR?

y+

U(y)

5O

NONLINEAR?

FIGURE layer

1.

and

A model

a linear

of wall-bounded

fluctuating

turbulence

outer

layer

driven

consist by the

of active active

non-linear

inner

layer.

with rij = -uiuj The

solution

of (1) assuming

transformation

in z, z and (U-

where

the

and

caret

denotes

The

basic only

this

of the

may

fluctuating

be found

quantity

number

applying

formula neglecting shear

Fourier

relating

near

the

From the

viscous

stress,

terms,

is related

the

velocity

S_ denotes U(y),

the

and the

spectrum prime

(6)

k_, denote

the

z-

c = w/k=.

that the

< uv

and

in the

driving

then

terms

region an outer

determine

are

outside

any

linear desired

stress,

>

to

(7) equations their

number for the

the nonlinear

flow

may

shear

component

wave

to the spectrum

are

that

of q one

quantities

s" = (k,/k) where

and

Reynolds

to the

mean that

and

q as a given

statistics

transform

model

wall

as the mean

quadratic

Fourier

stress

by considering

such

by applying

k= and

k, respectively,

r = -p By

quantity,

in the active-layer

region

flow field.

statistical

wave

Reynolds

hypotheses in a thin

layer

be determined

2 -k2)ZG=fT/ik=

a Fourier-transformed

of the

Calculation

large

may

yields

c)(_"-k_b)-U"f_-(1/ik=)(d2/dY

z-components

3.

q to be given t, which

(5)

and

using

transforms,

spectrum,

v-fluctuations

S_,

one

Parseval's finds,

for the

upon

Reynolds

through

+ (k=/k)= U"S , of v in a reference

denotes

differentiation

frame with

convected respect

(8) with

the

to y. From

mean (6),

it

Active follows

that

for small

viscosity

Layer

the

Model

299

v- fluctuations

may

receive

their

butions from the region near k_ -- O, in which case the second negligible, and with k = kz one finds after integration that

Sr= a result 4.

that may be shown

Application The basic

linearities the

to hypothesis

on which

are important

only

aid of the

turbulent tions, the

numerical

channel

although

the

terms Since

way

basic

flow

the

of (1)

results

shown

are

of the in Figs.

presented (arbitrary half width), which tenth

of the

Fig.

model,

3.

layer,

maximum

towards

distances

In Fig.

the

its main

k_ ). This

probably

tion.

The

however,

and

reflects

3h),

however,

from the

the nonlinear

of Reynolds

it receives

in ks at a higher elongation of the

resulting

from

contributions

from

value

this

of q is

shown in the

outside

the

(both

in wall

buffer

in k_ and

involved

in dissipa-

spectrum

the low

The

(h = channel of about one

flat

range

can therefore

spectrum

for k_, reflecting structures. The

is well

mechanism

of

field.

spectra

value than dominating

which

because

power

to be fairly

wave-number

cascading

stress

its main

high

data,

soundness

velocity

The

for

simulaquantities

square

channel.

base

accessible

root-mean-

are found

the q-spectrum responsible for the production more concentrated in the near-wall region. 5.

mean computed

with

computed

of the

the

of the

the wall

3g and

contribution

production since

the

statistical

spatial

2, the

center

from

at y+ = 76 (Figs. has

From

assessment

from

data

numerical

is not so easily

the

the non-

be tested

extensive

and appropriate

evolution

determined

buffer regions, with the cutoff dominance of high streamwise

spectrum

numbers.

that

may

scale). It has a maxinmm at around y/h = 0.1 corresponds to y+ = 18, and drops off to a value

3 for different

and the

hypothesis.

namely,

wall,

NASA-Ames

for a preliminary

2 and

the

A fairly

in the

be extracted temporal

was

mixing-length

is based,

near

simulations.

present

realization

model

region

generated

are stored,

hypothesis time

the present

low Reynolds

may

the detailed

data

of a single

to fairly

(9)

with Prandtl's

in a thin

been

in (8) will be

turbulence

turbulence has

limited

nonlinear

determined. of the

flow

term

contri-

U'S_,

to be consistent

channel

largest

is not large,

k_-end.

The part

be expected

of

to be even

Conclusion The

model

proposed

for

wall-bounded

turbulence,

namely,

driving of the turbulent fluctuations is concentrated the viscous and buffer regions, has been tentatively NASA-Ames

channel-flow

linear

are the

and

effects that

outside

nism

leading

time

realization;

semble

the

simulations.

strongest near-wall

to dissipation.

averages

near

The

region The

they

mean

for

a more

general

over

a large

number

in the examined

preliminary

the wall,

with

treatment, of realizations,

findings

primarily

spectra one

were would

the

nonlinear

near-wall region, in using data from the

a maximum

involve

and the

that

are that

the

at around the need

y+ = 20,

cascading

obtained

mecha-

from

to work

as well as to employ

non-

a single with

spectra

enin

300

M.

T. Landahl,

J. Kim

and P. R.

Spalart

1.0

q.5

0

,

,

,

,

I

0

,

,

,

I

,

.5

1.0

y/5

1.0

¸

(b) 0

i.,,I,.,,I

....

0

I,,,,I

20

....

I,_.,I,.,.I,,,,I

40

6O

80

+

Y Profiles

FIGURE

2.

a frame

of reference

this

model

stresses lation

will lead

and other making

region,

values

of q in Eqn.

convected

mean

velocity.

with

Reynolds

the

to a reasonably statistical of which

local

simple

quantities

use of a universal

the statistics

at modest

of the root-mean-square

model may

procedure

through

be found

If found

for determining

a comparatively

for the

nonlinear

from

(1).

numerical

the

simple

processes

to be sound,

in the

simulations

Reynolds

linear

calcu-

near-wall carried

out

numbers. REFERENCES

LANDAHL, 29,

M. T.

441.

1967

A wave-guide

model

for turbulent

shear

flow.

J. Fluid

Mech.

Active

Layer

Model

301

10-1 y/_ = 0.024 y+ = 4.37

y/8 = 0.024 y+ = 4.37

10-2

10-3

(a) 10-4

.L.,L.J

(b) ,hi,h|

....

I,,.,

I.,,,I,.,A--,I,

hhh|

....

I, ,,,I,,--I-J-J1p,;

....

I,,,,I,,,,h.,J..I,l,l.h|

....

I,,,,I,,,,I,.,J.J,I,hh]

, , I I I['"

10 o y/_ = 0.058 y+= 10.52

y/_ = 0.058 y+ = 10.52

10-1

10-2

(d) ,.,

100

, I,,,,I,,..I.,J,.I,h|,lJ

, , =,

I,,,.h,,,h.,J,.J,hhhl

101

3.

Spectra

of q at different

y-locations.

,I,,

102 kz

FIGURE

I ''

302

M.

T.

Landahl,

10 0

J.

Kim

and

P.

R.

Spalart

100 y/_ = 0.107 + y = 19.22

y/8 = 0.107 + y = 19.22

101

10-1

102

10-2

(e) 10-3

(f)

,.L....&.,I,I.I,I.I

I

'

' '

I...,I....I..&..I,hhh|

I

I

I

I

hi

"'h'''l='l'J'h

10-3

,

* * h,*,h,*,h*.JImi,hl,IJ

*

1 " ' I ,,,,I,.,.I,,.Ji,l,hl,hJ

I

I

I

I J*"

10-1

100 y/6 = 0.424 y+ = 76.35

y/_ = 0.424 + y = 76.35

10-1

10-2

10-2

(g) • ,..L.J.I.I.I.I

,

. ,

. h.*,l.,..l....l.J.hhLl

100

101 kx

FIGURE

3 ....

.

continued.

,

.

, h=,.l,,,J.,.,&,.A.h

10-3

(h) I

100

" ' I,..,I,..a.-,J,-I.I.hlJ

*

. ,

101 kz

* I ,.,,I..,,L...I,.J.l,l.hl

.

102

.

• • h.,

N88-23116 Center

for

Turbulence

Proceedings

of the

303

Research

Summer

Program

1987

Wave-growth turbulent spot By

D.

A kinematic waves

the

wave direct

1.

theory

at the

qualitative

breakdown

criterion

as the

and

1982;

oblique

to investigate spot

motions

wave

selection are

2 and

the

of turbulent

velocity

cause

in plane

procedure.

J.

Kim 3

of the

rapid

Poiseuille

is well The

in agreement

waves

(1987)

with

growth

flow. by

predicted

wave

those

values

of

It is found

described

of a Poiseuille

wave-packet before

extends

they

important

break

role in the

(Chambers This

poses

different

the

surrounding

the

into

rapid

Landahl's number,

obtained

in a

resolving

growth selection

of the waves mechanism

these

where

turbulence.

the This

the

Is the

waves just

induced

questions

1987)

wingtips,

i.e.

(T-S)

mode.

by the

et al.

1987)

waves

attain

of the

waves

seen

of the response

turbulence

the

high

do play

the present

work

addresses

spots.

The

layer

to play

such

spots

spots

caused

the

cause

by

in the

As a first

related

an

a role.

to disturbances

fluctuations?

observed in Poiseuille will also be considered.

wingtip

amplitude

waves

two

spot,

numerical

In boundary

not

spreading

a passive

of the

that

the

that

Henningson

sides

very

et

shown

A recent

shows

spot velocity

(Carlson

have

the

that

spot.

are

line

downstream.

indicates

growth

however,

questions.

flow

toward

on the

center

Experiments

& Alfredsson

(Henningson

spanwise

Are

laminar

1000.

into a turbulent

is the

as it propagates

waves

spot

1983),

following

mechanisms?

spot

flow spot

& Chambers

about

Tollmien-Schlichting

into down

UVL

is above

the

stable

, where

Henningson the

observed

flow can develop

Ucnh/v

1986;

around

least

Poiseuille

=

height)

et al.

develop

of the

simulation

( Re

half-channel

Alavyoon

consisted

in plane

number

& Alfredsson

step

of the rapid

question

of wave

Analysis The

wave

pattern

computations dimensional

1 Aeronautical 2

Landahl

of the wave

disturbance

Reynolds h is the

al.

all

T.

flow

Introduction A localized

the

wingtip

phase

1, M.

is used

behavior

angle, and simulation.

if the

2.

Henningson

wave

observed

that

S.

associated with in plane Poiseuille

Massachussets

3 NASA

Ames

to be analyzed can

be

quantities Research Institute Research

found are

Institute

non-dimensionalized of Sweden

of Technology Center

is shown in Henningson

(FFA)

in Fig. et

1.

al. by the

Detailed (1987). centerline

descriptions In

what velocity,

of

follows, UCL,

304 and

D. S. Henningson, the

channel

spanwise

half-width,

directions,

vertical

velocity

a distance

of about

shows

modal

the

lc shows waves

the

i.e.

the

from

....

wave.

_'L

The

(¢) between

where

the wave

the

streamwise,

la shows

for the

spanwise _L'11

"

the

vertical

A plausible

(Witham

calculations

of the

part

point

at

first

few

this

time,

maxima

at the

as they

J. 11-__.

by I-lenningson

_f

velocity

vector

and

exceeds

explanation

packet

the

wave

(1987)

streamwise

a, a,/3

growth

to analyze of the

to have

into

the

1.

have

•

shown

that

about

-65 °. The

0.7 in the streamwise here is taken to be

seen

at

the

their

wingtip time and kinematic

interaction.

The

is interaction spatial wave

starting

scales theory point

of

form, 0 = olz

and w are assumed

are assumed

,

direction

and the waves. If the are widely separated,

ae i0

The waves

1

The

0.02.)

of the

is appropriate

is a wave

lb Fig.

centerline.

•

et al.

Fig. and

propagate

Jl

of the

has travelled

t = 258.

amplitude

variation

a._'_11--

spot

and

direction of the waves are approximately 0.6, the vector (k = V/a 2 +/32 ) about 1.8, and the angle

changing mean profile motion and the waves

1974)

theory

where

of + 0.01 contours

front

of magnitude

_ .....

vertical

a top view

amplitude

an order

....

number

The

of the spot interface at the wingtip is approximately and 0.12 in the spanwise direction. (The interface

between the of the mean

the

z, y, and z denote

waves

the phase speed in the streamwise absolute value of the wave number velocity direction

J. Kim

its generation

almost

1_1

and

centerline.

of the

to grow

a.t_."

T-S

200

shape

T. Landahl

Fig.

channel

corresponding

are seen ___1-:1_

h, and

respectively.

at the

M.

+/3z

to be slowly a known

-- wt

varying

dispersion

functions

of space

and time.

relation

= which

relates

components. be shown

the

variation

of the

Using the definition (Witham

angular

of the phase,

the

OW

dt

Oz

a/3

ow

dt do.,

Oz OW

dt 1 dA

Ot 00W

rays defined

00W

Oa

A dt the

0, and

to

1974), da

along

frequency

by dz

OW

dt dz

Oa OW

dt

013

Oz 0/3

the

wavenumber

dispersion

relation,

vector it can

turbulent

spot

305

-39.270 0.23 -31.416 -23.562 Z -15.708 -7.854 (a) 0 117.69

i

i

133.34

l

148.99

164.64

180 29

i

195.94

211.59

227 24

X

1.0

0.25

0.6

3

h

0.2 Y

/2

-0.2

1 V

,..

0 2

-0.6 4 (b)

-1.0

-0.25 0 V

-0.25

FIGURE

1.

centerline:

x=186

where

-----, and

A is the amplitude.

group

velocity, T-S

two-dimensional density

--, z=186,

(b)

Profiles

x=186, z=-25.5.

of the vertical

of the

z=-29.8;

....

vertical

-7.854

velocity velocity

, z=186,

(c) Spanwise

along

wave The

action wave

density,

variations

_'9 = (aW/Oct, waves wave.

riding

which

properties

are

0

at the channel for the

z=-28.2; of the

OW/O_).

is proportional

seen

to vary

Landahl

on an inhomogeniety

........

vertical

first

few

, z=186, velocity

at

He was

able

to

along

(1972)

to integrate

the

square

of the

rays given

by the

the

above

theory

of a larger

scale

locally

for the

wave

action

normal

to the larger

applied

consisting

the

the

equation

a ray to yield A Ao

where

-23.562

y=0.

wave

to oblique

of + 0.01 contours

Re=lS00.

maxima:

z=-26.7;

-39.270

Z

A top view

(a)

t=258,

amplitude

0.25

(c)

cgn is the group

velocity

1 Cgn

of the small

--

C0

scale

oblique

waves

306

D. S. Henningson,

two-dimensional Notice

wave,

when

of wave

wave,

energy

their

the

Co is the

cg_ approaches

is a result by

and

co the

energy

and

phase

velocity

of the

amplitude

When

the

piles up on top of the large

incoming

ix

T. Landahl

wave

focusing.

waves

but

no energy

amplitude of the oblique waves. As a first try we shall assume fOllnC]_

M.

that

lnr'_

twn-d

1972):

i.e., we will examine

J. Kim two-dimensional

will increase oblique

wave.

dramatically.

waves

approach

one

since

energy

is constantly

can

leave,

this

results

in an increase

the

edge

_,'nA

of the

'_ ......

spot,

where

1"_A_1-.,1'_

÷ko

larger

supplied

the

L.^^1._1

This

the

in the

waves

.......

are

:

---'--

i

(Landahl any

combination

the

wingtip.

fl and

of wave If the

0;, then

edge.

Thus,

The

wave

velocity

criterion

both

relation

Orr-Sommerfeld

and

for the

equation

U and

tively.

U and

W are the

where

Co, c2 and 2 using

are

obtained

la

indicates

are found

wingtip

waves,

for

edge

velocity

at

to the

combination

as it approaches and

selection

by solving

et al 1087).

of a, the

of the

spot waves.

an extended

The

form

appropriate

mean

c4 are

from

the

the

fitted

velocity

the

inflectional

position

profiles

This

dam

(1972)

used

term

in the

equation

growth/decay short times (Landahl relation

the

of the

requires

components,

the

required

fitted

to the

rates and that only. Assuming

Results

and from

at the

combination

one

la;

and

the

becomes

large

become

greater.

velocity,

which

is solved,

the

which action

group

in Fig.

position

is

velocity

the development this to be true

adding and

the

the

in Fig.

is in the

departure

This

results larger

of from

weight found

theory.

of the This

real

is

Lan-

a growth/decay use

in

They

line

arrow

wave

is calculated.

of the wave the breakdown

the

relation

kinematic

involves

density

seen

figure).

have

dispersion

of the

velcities

in the

mean

a modification

the

wingtip

angle

parabolic

wave

are

rate part

requires

packet is considered criterion is still

1982). In the following we will thus use the real part in our effort to look for waves that satisfy the breakdown

Results

respec-

4)

¢ = 65" is shown

at this

wave

spanwise

approach,

when

2 -Jr- c4y

at the

edge

equation

for the

(only

as the

from

a simple

relation

spot

that

for higher wave angles. When the Orr-Sommerfeld

dispersion

velocity

and

Profiles

angle

indicated

of the

Note

-4- c2y

constants. wave

character

complex.

spanwise

averaging

-- y2)(Co

fitting

tangent

of motion.

usually

and

by horizontal

the appropriate the

+ W(y)tan(¢)

parabola

direction

cated

growth

for growth

waves

streamwise

(1

3.

for a specific

(see Henningson

mean

W are found

to a modified

the

is satisfied a rapid

T-S

of the

is

where

Fig.

velocity is equal

a mechanism

U(y)

fitted

group

frequency,

will experience

this provides

dispersion

of the

breakdown

the

if the

numbers

of

small for valid

of the dispersion criterion.

Discussion the

wingtip

solution in Fig.

of the

Orr-Sommerfeld

la can

be seen

in Figs.

equation 3-5.

for the Contours

position

indi-

of the

group

turbulent

spot

307

1.0 0.8 0.6 0.4 0.2

y

)

0

\

-0.2 -0.4 --0.6

"

_

STREAMWlSE

-0.8 -1.0 -0.2

-0.1

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

1.3

U FIGURE

2.

Mean

velocity

profiles

at z=186,

z=-30

and

_b = -65

°.

velocity perpendicular to the spot edge (i.e. in the direction of the arrow in Fig. la) are plotted as a function of k and _b in Fig. 3. Figs. 4 and 5 show contours of the phase

velocity

relation.

in the

Note

that

_b = 45 ° (Fig. The

3).

breakdown

streamwise the

group

The edge

criterion

The

waves

phase

velocity.

time the

ment

be noted

cannot

behavior values,

is able

It should

the

part

from

to large

the

fulfilled

are damped

wave

at this

normal

However,

into

higher This

wave

to the

wave

position

(see

a wave

using

growth, effort the

spot

with the

and

its

in a future

study

and calculating

ridge

going

from

wave

parameters.

that

exact

the

and

the

k=1.7

the

through

the

amplitude

along

agree-

observed with

wave full

and

qualitative

the

together

observed

to attack the

5) and

observed

criterion

explain

Fig.

and

around

going

breakdown thus

angle

At

to see that

ridge

of the

we might expect and at the same

the

analysis

it is encouraging

velocity,

0.23.

for k = 1.0 and

number,

numbers.

is close

was

at the wingtip

we follow

this is an approximate

group

to select

rays

to grow.

that

of exponential wave

towards

a wave

dispersion

simulation

amplitude

higher

of the

of 0.25 for k = 1.0 and

to be approximately grow

To find such

start

here

be an worthwhile the

imaginary

location

has both

waves

3 up

be expected.

of the

requirement tracing

in Fig.

°, the waves

It should

seen

observed

the

its maximum

at this

should

exponentially.

value

_b = -65

actually

and

attains

exponential growth for higher wave angles. Thus that approximately fulfills the breakdown criterion

is gowing peak

wave

However,

only experience to find a wave

velocity

is thus

_b = 45 °. This particular spot.

direction velocity

the

motions.

problem them.

by

308

D. S. Henningson,

M.

T. Landahl

and

J. Kim

2.4 2•2

•

!

0

-10

FIGURE the

3.

Contours

direction

of the

Finally, outside

the the

cussed

-20

of the

that

are

first

by for example

wave

energy

velocity

group

arrow in Fig.

picture

spot

-30

velocity

emerges

from

generated

mechanism

-50

-60

perpendicular

to the

-70

spot

edge

-80

(i.e.

in

la).

Li & Widnall

focusing

-40 ¢

and

the

present

by the

moving

(1987),

then

analysis

is that

turbulent they

the inflextional

the

waves

disturbance,

experience

character

as dis-

a growth

of the

by the

effective

mean

profile. REFERENCES

F.

ALAVYOON, in plane

D.

R.

W.

Poiseuille

CARLSON,

study F.

D. S. HENNINGSON,

and growth• D.

spots D.

S. E.

A.

Phys•

in plane

S.

Fluids•

_5 P.

H.

Poiseuille

S. HENNINGSON, turbulent

WIDNALL, in plane

_

S. HENNINGSON

spots

P.

H. ALFREDSSON

flow - flow visualization.

of transition

CHAMBERS

_

P.

W. 26,

R.

M.

F.

CHAMBERS

Fluids

PEETERS

flow•

1983

. 29,

1982

J. Fluid

Turbulent 1328-1331.

A flow-visualization

Mech••

Turbulent

spots

121,487-505. spots,

wave

packets

1160-1162.

ALFREDSSON flow•

in plane

_

Poiseuille

Phys.

1986

J. Fluid

SPALART

_

1987

The

wave

structure

Mech.•

178,

405-421•

J.

1987

Numerical

KIM

Poiseuille

and boundary

mechanics

of breakdown.

layer

flows•

of turbulent

simulations Phys.

Fluids.

of 30,

2914-2917. M. T.

LANDAHL

1972

Wave

J. Fluid

Mech•.

56,

775-802.

turbulent

spot

309

/

2.4 2.2

/ /

OAO

.

.

-60

-70

2.0 1.8

1.6

0.35

1,4

0.30 1.2

1.0 0

-10

FIGURE

4.

-20

Contours

-30

of the phase

-40

velocity

-50

in the

streamwise

-80

direction.

2.4 2.2

2.0 1.8 k 1.6

1.4

1.2

1.0 0

-10

FIGURE

5.

-20

Contours

-30

of the imaginary

-40

part

-50

-60

of the dispersion

-70

relation.

-8O

310 M.

D. S. Henningson, T. and

F.

LI

LANDAHL packets

with

B. WHITHAM

The

small

& S. E WIDNALL

concentrated G.

1982

application

1987

Linear

Wave

T. Landahl

Phys. patterns

Submitted and

and

of kinematic

dissipation.

disturbances. 1974

M.

nonlinear

wave

Fluids.

25,

in plane

to J. Fluid waves.

J. Kim theory

trains

1512-1516.

Poiseuille Mech.

Wiley.

to wave

flow

created

by

Center for Proceedings

Turbulence Research of the Summer Program

311 1987

Appendix During

the

structures

was

in turbulence Several

modeling

to explore

participants position

organizers

the

problem views

ideas were

papers

that

the

authors.

by

the

modeling.

a special

half day workshop

was organized in the

use

by Prof.

of the

knowledge

on the

role of coherent

A. K. M. F. Hussain. of organized

The

structures

modeling.

as submitted

the

program

in turbulence

objective

written

summer

of the

Summer

of incorporating This

reflects

expressed

asked

here

the

to make

are included While

these

Program the great

presentations,

feel

coherent difficulty

will be helpful

and

in this appendix. contain that

some

research

of achieving

towards

this

end.

this

prepared

statements

interesting

no solution

structure

five of these

These

was

appear

observations, put

forward

in Reynolds integration.

to

stress Perhaps

N88-2311 Center

for

Turbulence

Proceedings

of the

Research

Summer

313

Program

1987

Coherent By

Our

mental

molecular as

to

on

picture

motion

whether

the

to

there

strength

belief

has

been

(1883)

•

Prandtl

•

surface flow Theodorsen

•

Townsend

•

Grant

•

Bradshaw

•

Crow mixing

•

Brown & Roshko (1971) 2D vortex roll structure

(1956)

length

- WEAK

et al.

(1964)

&Champagne layer.

1964

and

1971

eddies from the norm, at least for discussion

the

orderly

Now

main

are

the

time

scale

energy

transfer

coupling

(but

large

gets

was

3D)

large

eddies

grew

or unusually

from

Ahlborn

dissipation

in

the last

picture,

part

the rest.

with

structures!

lived,

or unusually

research

- and unlike

is that

of course

the

rate)

or

it clear

that

rolls observed representative

a mixing

layer

of the This

According

disturbed

shear

stress?

result

was

large

to be controlled turbulent energy). "Weakness"

an eddy

viscosity

1

The How

eddy

equilibrium

of the for

large the

eddies

smaller,

College,

London

therefore principle fraction

hypothesis,

was

required

energy-containing

a two-

is,

layer

of turbulence.

an

rate)

unusually

dimensional

is

an unusually small rate of weak

eddy

may

in the vortexstretching simulations of Metcalfe et

conditions

can

$1are we studying

have

a large-eddy

are chasing the right, any turbulent flow is

be studying the eddies that carry, that presentations of educed eddy of total shear stress that is carried turbulent

mixing

for

the

"equilibrium"

eddies.

Grant

wakes did contain a large fraction - but the concept of large eddies ignored for the next 15 years.

main questions about "orderly structures" different are they in different shear layers?

Imperial

/ orderly

mixing

(i.e.,

shear

stress)

by large eddies (more or less filling the shear layer but not carrying This is a paradox - how can eddies be weak in energy, strong in

large eddies in boundary layers and This killed the equilibrium hypothesis mixing should not have been largely (i)

initial

eddies

in the picture

- that

that

large

eddies. Thus an unusually

turbulence

suggests

by Brown & Roshko and others. We need to make eddies! A related point is that our main object in

to Townsend's

latter

but finite vortex rolls swept back at various angles, presumably strongly and therefore having shorter lives than the parallel two- dimensional

to improve predictions of shear stress, and we should or influence, the shear stress. It seems a good general shapes should be accompanied by an estimate of the by eddies of that shape.

was supposed much of the

the

the

two-dimensional

TKE/(production

with

make

in

that

Unless

that

traditional

al.

consisting of long with each other

structure

implication

orderly

it does not share so directly eddies do. The mixing-layer

(1987)

the

long-

be exceptionally long- lived, simply because "cascadse" of energy as threedimensional structure interacting

layer. orderly

into

15 years'

thought

in that

and

of water-

tails.

mixing

Re-dependent?)

energy)/(dissipation

rate) eddy

method

viscosity

surface).

correlation

energy-containing, shear-stress-carrying from flow visualization, say) implies

the long-lived

the

eddy

change from Townsend's more descriptive away from the inner layer of a wall flow.

two-dimensional

kinetic

of the deduced

to

the

question in general

wake.

eddies

(but

structures is instructive.

near

it mimics

is some

leading

from

and

that

there

history

by the

deduced layer

2D

previously

essentially

(turbulent

(small

between

one than

are

shapes

STRONG

strong,

the

motion

it is perhaps a confusing of the dominant eddies

impression

always a representative long eddy lifetime (as

STRONG

unusually

longer-lived

structures

with

at

theory,

inspired

in boundary

- VERY

Townsend's

large label,

the

eddies

hypothesis

of coherent

look

kinetic

- confirmed Crow g_ Champagne may not be very Re-dependent.

is used

structures

A brief with

the

though

over-selling

(apparently

eddies

- VERY

to distinguish

from

organized

forces two-dimensional of assorted sizes.

large

(1971)

an

layers. analogy

"lumps"

large

term

Certainly,

ball"

i.e.,

x

progressed

is highly

overshoot,

in mixing

- STRONG

has

turbulence

visualization, which (1952) - horseshoes

(1958)

Between

an

- "billiard

-mixing

BRADSHAW

structure

that

of experiences

Boussinesq concept.

Structure

PETER

of turbulence the

•

(1925)

7 ".

or "large

eddies"

are:

hypothesis. (1958)

showed

The that

of the turbulent energy. as large contributors to

314

P.

Bvadshaw

(ii)

Can we usefully combine a model of the orderly structure with a cruder representation of the small-scale, less-well-ordered eddies as in Townsend's large- eddy equilibrium hypothesis or the Murthy &Hong Large Eddy Interaction Model? (iii) What about the all-important boundary layer? (Especially the outer layer - engineers think they know all they want to about the inner layer.) Suggested answers: (i) Large "orderly structure" is just an extreme case of Townsend's large eddies, which were always envisaged as different in different flows. Therefore study of orderly structure is not a route to a universal model (at least at Reynolds-averaged level). The oersistenc_ nr orderly __*.ruc_.'-'-rcit, _:riticai to the success ot "'zonal average" modeling - if large eddies/orderly structure persist for a long time after a change of zone, the change of empirical coefficients at the edge of a zone will have to be governed by a complicated rate equation. (ii) Except in the mixing layer and the inner layer of a wall flow, the background turbulence carries a very significant part of h--6- but maybe we can manage with just a few modes (not necessarily superposable Fourier modes). (iii) In the outer part of a boundary layer, the large eddies/structures seem to be not-very-wellordered eruptions, probably horseshoeor hairpin-like (beware low- Reynolds-number simulations), because the outer structure depends on viscosity up to Ree = 5000 (as shown by wake parameter behaviour - see also Murlis et al., 3. Fluid Mech. 122, 13, 1982). Since the outer part of the flow presents the biggest challenge to turbulence modeling (provisionally accepting the engineer's view of the inner layer!) study of the more spectacular forms of orderly structure is unlikely to help us with the boundary layer problem. Of course, study of the inner layer is helpful to basic understanding: at least the scaling in the logarithmic region is simpler than usual even if the turbulence is not. As a final comment, it is still difficult to see how conditionally-sampled data on orderly structures, whether from simulations or from experiments, can actually be used in Reynolds-averaged models, although qualitative understanding is still valuable. However, if we could derive exact transport equations based on some compromise between Reynolds averaging and solution of the time-dependent Navier Stokes equations, we would have a considerably better chance of using our knowledge of orderly structures.

N88-23118 Center

for

Turbulence

Proceedings

of the

Research Summer

Coherent

315

Program

1987

Structures-Comments By

1.

Introductory There

is

now

vary

but

overwhelming

there exist "structure."

the

J.

C.

It.

HUNT

on

Mechanisms

flows,

despite

1

remarks

ing random, characteristic large-scale through

!

distributions flow and

the

evidence

regions These

that

moving regions

of velocity and/or interact with other

distributions

in

most

with the are called

turbulent

vorticity remain coherent even structures. In most flows the

of velocity/vorticity

the

flow where the velocity and "coherent structures" because

remain

similar.

There

as these sizes of

is also

flow

field

be-

vorticity have a within them the structures move these structures

evidence

that

there

is a significant degree of similarity between these distributions in different flows (Hussain 1986). Since flow enters and leaves the bounding surfaces of these structures, a useful definition, following Hussain (1986), for coherent structures is that they are "open volumes with distinctive large-scale vorticity distributions." The the

following

study

schematic

remarks

dynamics

of coherent

of the

been concentrated move to examine described here.

2.

The

on the measurement the dynamics of CS,

origins

Coherent

of

coherent

structures

that

non-uniform

some

small flow.

small

arise

by

flows

are

disturbances

in amplitude Even as the

the original disturbance, Consequently

two

CS).

but their amplitude

linear

A general

feature 1987).

vortices

of the

at

high

main

types

(primary at high field

or secondary) Reynolds

number

theory

amplify.

structure 1).

remains

The

they

most

are

usually

remain

unstable

unstable,

similar

of considerable

disturbances

so

scale is usually of the same order as that of the original mean disturbances grow to the same order as that of the velocity of

in

parallel 1986).

to

of the

localized

the

primary regions

examples

are

vortices

to that

value

in the

both and

of

the

in analyzing

may

original CS

be

small

and

in esti-

on

amplification homogeneous the

non-linear

of Cambridge

flow

and

and

secondary

the

flow

rollers

(based to have

that

the

instabilities

as shown and

link

braids flows.

on global or characteristics

by

primary

is that

recent

in free These

"roller"

they

lead

non-linear

shear

layers,

regions

to

studies and

the

of concentrated

local scales) just great enough similar to those in coherent

numbers.

isotropic

rotates

mean

in boundary-layer

Reynolds numbers are usually found

Selective

possible

Computations,

University

there is now a welcome A few of them will be

of mechanism.

computer,

of disturbances turbulent e.g.,

wind tunnel, e.g., Champagne et al. 1970; it may at the wall or at the outer interface in the boundary

1

has

length of the

"horseshoe"

an initially

is certainly

stretches

for

to-date

velocity

growth

Reynolds (ii)

shear

directions

research

to the

Well-known and

vorticity, which occur at for turbulence to persist,

Consider

fruitful

CS

and kinematical analysis of CS; by a variety of different methods.

Instability

stability

of vorticity

Smith

structures

possible Most

consequences, e.g., their effects on noise generation (e.g., Gaster et al. 1985). primary instabilities have grown secondary instabilities develop, often with vorticity perpendicular to that of the primary instability, for example the "braids" that form

a concentration

(This

about

(hereafter

generated

in free shear layers with vorticity vorticities (e.g., Bernal & Roshko

spanwise

statement

structures

flow, the length scale and flow as in the free shear layer (Fig.

mating their Once the in directions

(e.g.,

a personal

structures

(i) When

are

by Rogallo

the vorticity

in such

(1981),

by

velocity

the

mean

field

field

introduced

and

approximately

into

Rogallo

1981,

broadly layer,

approximate the turbulence e.g., Townsend 1970.)

and

a way that

linear

by

the eddies

Lee et al.

(1987),

become

elongated

a shear

flow.

possible

in

generated show and

that

the

develop

a

816

J.

C.

R.

Hunt

SECONDARY "CENTRI FUGAL"

INSTABILITIES

I

PRIMARY ROLLS SHEAR INSTABILITY

"BRAID" SUB-STRUCTURES

COHERENT

STRUCTURES

FIGURE 1. Coherent structures generated from instabilities: Primary and secondary instabilities in a free shear layer leading to two kinds of coherent structure -- "rolls" and "braids." The latter are also known as "sub-structures."

sharper boundaries (Fig. 2). This is the explanation of the evolution of the velocity spectrum in Rogallo's computations from an exponential decay (indicating very smooth velocity distributions in the eddies) to a k-2 decay (indicating discontinuities on the large scale). Linear calculations using Rapid Distortion Theory (RDT) also show these effects; similar coherent structures are found as in the non-linear simulations. (The analysis of the RDT spectra needs to be done.) 3.

Interactions

between

a coherent

structure

and

After the formation of a CS, its subsequent existence surrounding flow. This varies considerably between flows.

its

surrounding

depends

flow

on how it interacts

with

the

(i) u.ifo,.,. /tow, Although CS are not generated in uniform flows, they are often moved by the flow or propagate themselves into regions of approximately uniform velocity; CS in wakes far from the body approximate to this case. The simplest type of CS to consider in this situation is the vortex ring or the vortex pair (Maxworthy 1977). These CS preserve their structure, but they always lose some vorticity as they propagate through the flow (Fig. 3a).

Coherent LOW REYNOLDS

NUMBER

HOMOGENEOUS

ISOTROPIC

Structures-

TURBULENCE

Comments

317

log E(k)

log k

AFTER

STRONG

SHEAR

dUt >>1 /3 = dy

_/

log E(k)

SHARP

GRADIENTS log k

FIGURE a typical (Rogallo,

2. Coherent structures generated by deforming a random eddy or CS as shear is applied to homogeneous turbulence. 1981; Lee et al., 1987)

(ii)

Neighboring

structure8

In free shear layers, wakes, and jets the CS are formed in thin layers the surrounding flow on the CS must be caused by adjacent or nearly different to boundary layers where each CS is surrounded by other CS One could characterize these interactions as 1 + 1 + 1 + .... These be idealized by considering the interactions between pairs of vortices. useful conservation conditions (Aref 1983), and insights into whether other or merge or pair (e.g., Kiya et al. 1986). (iii)

Many

structures

velocity field. Distortion of Note the change in spectra

or a mean

so that the main effect of adjacent CS. This is quite (Fig. 3b). kinds of interactions can Such studies have yielded the vortices go round each

fieldf

In turbulent boundary layers each CS is surrounded by others, and also by small-scale incoherent motion. In such cases it is no longer profitable to examine each interaction between the structures; it is more practical to represent the whole velocity field as that of the CS, U, plus a mean velocity field, _, induced by the vorticity within all the CS, and a random component, at I to represent on the large scale the random locations of the CS, and on the small scale the incoherent motions (Fig. 3c). In other words, 1 + cc + _l, is equivalent to U + u + u I. At present we understand little about the mechanics of the interaction between a finite volume of fluid containing vorticity and the surrounding flow. Most research on the dynamics of turbulence has been focused on the interaction between small perturbations and various kinds of mean flow, e.g., Townsend (1976) or Landahl & Mollo-Christiansen (1988). This is perhaps surprising since Prandtl's (1925) mixing length theory is based on a qualitative model of how "lumps of fluid" -FKissigkeit ballen -- interact with a shear flow.

318

J.

C. R.

tfunt

(a)

l _

PAIRING

(b)

PASSING

FIGURE 3. (b) Interaction

O

(a) A ring vortex in a uniform between a Coherent Structure

TEARING

flow showing slow shedding of vorticity (CS) and neighboring CS (1+1 ---).

downstream.

Some general questions about this problem were set out by Hunt (1987): i) What is the applied force and velocity of a volume V in a non-uniform unsteady flow with velocity u? For inviscid flow (a good idealization for a short time), analytical expressions have recently been obtained for spherical or cylindrical volumes moving in shearing or uniformly straining flows. The initial movements of a lump-or CS- when displaced in a shear flow are largely controlled by inertial forces, in particular the added mass and lift forces. One finds that transverse displacements generate streamwise velocities of the lumps and thence Reynolds stresses-the CS analogy of the small perturbation Reynolds stress found in RDT or second-order calculations (Fig. 3d). (ii) How does the volume deform as it is accelerated when displaced across a shear flow? Flow visualization of the eddies or CS in a turbulent boundary layer show that as they are ejected from the wall form into "mushroom" shapes. These are very similar to the shapes of vortex sheets surrounding volumes of fluid which are suddenly introduced into jets (Coelho g_ Hunt 1987). In both cases the vortex sheets roll up into concentrated vortices characteristic of the mushroom

Coherent

Structures-Comments

319

+

(d)

i U,cs oy o (y)

l

=/

I _/o,,: /

FIGURE

3.

(c)

Interaction

with a mean flow u and leading to the generation a shear

flow

(Hunt

1987).

between

turbulence of Reynolds

a CS u e. (d) stress,

and

VELOCITY

OF THE CS

surrounding

CS

Movement -u_v _ --

--

of a volume an idealization

equivalent

to the

of fluid across of the mechanics

interaction a shear flow of a CS in

320

J.

structures. features

These of CS.

The other distribution certainly

4.

approach in a CS

led

many

such

Since

to and

to some

to have

analyzing to compute

interesting

the

forms

insights

(Moin

general

of

structures

on

coherent

coherent

structures

them.

field.

Because

a significant

are

regions

Consequently, they

proportion

proportion depends To compute the

of the

on the velocity

experiments, beginning

and detailed analysis of CS to make this possible (Hussain examples

of dynamical

vorticity the

of the

caused

by

relative

around distortion

bluff bodies and of the vorticity

internal

straining

motions

process

has

somewhat

been

CS

can

depends were

5.

detail

in a cross

at high similar

another

Concluding

Remarks

suggestions

for

difference

in

However,

useful

statistics

the

actual

velocity/vorticity here.

I am summer

to and

for

reflects. I am also grateful John Kim, who gave me analyze

the

AREF,

H.

J.

1983

stress.

The

for its

all

the

"life."

shear

regions

flows,

are

of the

around

to the

key

Recent

(i) amplification

the

CS,

amplification

CS.

These

are

(ii) the

particularly

namely

the

vortex

& Hunt

ring,

1987).

shed vorticity produces is another mechanism

free

actual

the spreading of the vorticity of into the "wake" of the CS. This Coelho

in the

in the

ve-

CS contribute

account

is similar

of the

the This

them

of the

1986; Hunt 1973) (Fig. 4); turbulence within the CS, by

of CS,

that

at

this

stage

interpreting density

it

but the

Hussain

is not

surrounding

shear

the

and For

the

those

an unstable whereby one

flow.

layer

This

essentially

secondary

to the organizers some interesting

1987

Lecture

Ann.

Rev.

at CTR

Fluid

Mech.

not

to

to

the

me

how

in

fact

structures

follow

by Prof. is perhaps

the

the

on

School. 345-

for

seminar

of the summer other perspectives

Summer

389.

dynamics

of velocity

is necessary

discussions

15,

CS

detailed

sampled velocity measurements This approach may lead to the

which

for organizing

of

of Houston Ames. There

measurements

CS,

interesting

dynamics

so clear

probably

sampled

many

the

conditionally distributions.

flow, in

into

at the University Stanford/NASA

REFERENCES

R.

Reynolds

are:

This

within

the

straining

controlling of vorticity type

both

components

of turbulent

and

the boundary

them.

ADRIAN,

to conduct

of motion

considered

flow.

turbulence

whereas

field

CS during

simulations

number body.

of investigation

of the

Fazle

Reynolds of a bluff

being followed Research at

distributions

school,

and

the

of the

(Maxworthy,1987;

small-scale

conditionally

grateful

strength

to be

simplest

flow

initially,

topics

the CS will emerge from conditional joint probability certain

the

of the vorticity flow. That has

necessary

fluctuating

and

stagnation

on

necessary

processes.

the lines of study that are at the Center for Turbulence significant

and

process

by inviscid

for

enough to that

CS

a diffusion

layers

be

(cf. Adrian 1987). it is necessary to

the

need

CS

will

scales

turbulence,

at the

it

a velocity

the

energetic

of the CS, i.e., the detrainment

in

produced

These

shear

jet

produce on

the

the growth control (iii)

studied

CS it was fond that velocity distribution

induce

in direct 1987).

that the

complex

more

found et al.

But

and

most

and

flow

between

they

of the CS by the CS

processes

and

of the

them.

cross jets in turbulent flows (Hussain and amplification of the "incoherent"

important for determining the CS. They also largely

is one

flow

mean

of the

movement

surrounding

motion

of course

1986).

whole

and

energy

the

al.

from

the

definition affected

processes

small-scale

affect

the

largest

total

precise fields

et

concepts

affect

dynamical

Three

that

the

persist-which

of vorticity

they

contain

that

dynamics of CS is to take a typical form its development in an appropriate shear

to obtain

outside

locity

appear

computations

Effects

and

vortices

C. R. Hunt

on

coherent

the

help

dynamical

coherent

of

in terms prediction

of of

structures.

in elucidating studies

structures

structures,

which

of

mechanics

of individual will

many

Hussain and one area of

at

the

outlined

the this

CTR paper

school, in particular Parviz Moin and on coherent structures and how to

Coherent

321

Structures-Comments

INCIDENT TURBULENCE

(I) STRETCHING OF VORTIClTY OF INCIDENT TURBULENCE

(II) SHED VORTIClTY FROM CS

FIGURE 4. Effect of a CS on the surrounding flow in high Reynolds number turbulence. (I) Amplification of turbulence in the surrounding flow. (II) Shedding vorticity which leads to further turbulence in the surroundings.

L. P.

BERNAL,

CHAMPAGNE, COELHO, cross

& ROSHKO, F H.,

HARRIS,

1986

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_

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J.

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E.,

C.

R.

1973

C.

R.

1987

HUSSAIN,

A.

M.

F.

HUSSAIN,

A.

M. F.,

M.,

_

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_

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Verlag,

N88-2311 Center for Proceedings

323

Turbulence Research of the Summer Program

Coherent

1987

Structures By

9

and JAVIER

Dynamical

JIMENEZ

Systems

I

Any flow of a viscous fluid has a finite number of degrees of freedom, and can therefore be seen as a dynamical system. For a turbulent flow, an upper bound to this number was given by Landau & Lifshitz (1959) and scales as Re 9/4, which is usually a rather large number. Lower bounds have been computed for some particular turbulent flows, but also tend to be large. In this context, we can think of a coherent structure as a lower dimensional manifold in whose neighborhood the dynamical system spends a substantial fraction of its time. If such a manifold exists, and if its dimensionality is substantially lower than that of the full flow, it is conceivable that the flow could be described in terms of the reduced set of degrees of freedom, and that such a description would be simpler than one in which the existence of structure had not been recognized. As a trivial example, consider a particular two-dimensional flow for which we can prove that, after some time, most of the vorticity concentrates in a few compact vortices. Such a flow could be described by a few differential equations, and, presumably, easily integrated. Homogeneous, two- dimensional decaying turbulence seems to follow roughly this model (Benzi et at. 1987). Other examples of the same type are transitional Taylor- Couette flow and Rayleigh-Benard convection. After the initial loss of stability, both systems develop attractors, different from the initial equilibrium, and that can be described in terms of structures (rolls or cells). Although the initial bifurcate states of those flows can hardly be called turbulent, the secondary bifurcations that grow from them can be analyzed, at least for a while, as small perturbations of the initial structures, usually described in terms of their positions and intensities. Technically, we speak of a projection on a central manifold (Demay & Iooss 1984); physically we are talking about describing a turbulent flow in terms of a few degrees of freedom. Another recent example of the same situation is the appearance of disordered states in two-dimensional Poiseuille flow, starting from bifurcations of nonlinear trains of Tollmien Schlichting waves (Jimenez 1987). The common feature of all these flows is the existence of stable attractors, whose dimensionality is much lower than that of the full flow, and towards which the flow tends after some time. Under those conditions, the flow can be described, up to some level, by the properties of the attractor. Most "attractors" found in nature, are, however, not stable, and cannot be really called attractors at all. The system will approach them for a while, only to be repelled once it gets near the central manifold. The simplest example of this behavior is the linear differential equation ytt + sin_! = 0, which represents a circular pendulum. If the system is given proper initial conditions it will approach the position at which the pendulum is pointing "upwards," spend some time near it, and fall back to make a quick revolution across the lower part of its trajectory. Even in this case, the system expends most of its time in the neighborhood of its top (unstable) equilibrium point, and can be described approximately as a being in equilibrium at that position, together with some model for the fast motion in the lower part of its orbit. Perhaps the best example of this situation, in a flow, is the plane temporal mixing layer. Here the "attractor" is a uniform row of compact vortices, and the flow quickly tends towards it. But this solution is itself unstable (mainly through pairing), and is eventually abandoned by the flow, only to converge to a different solution of the same kind. Even so, a model of the flow as a uniform vortex row, with a suitable approximation to the "sudden" pairing process, has been shown to give rough approximations to quantities such as spreading rates (Jimenez 1980) and concentration distributions (Hernan & Jimenez 1982). A more careful perturbation analysis on the lines outlined above has not been attempted, but might be expected to give more accurate results. Other turbulent flows have phase space structures which are presumably more complicated. The next best plausible candidate for eduction of the complexity of a turbulent flow are the sublayer ejections in wall-bounded turbulence. Recent observations (Jimenez et al. 1987) suggest that the basic structure in that flow is a self-reproducing ejection, that could perhaps be described as an unstable limit cycle. It is not clear, at present, how to treat a dynamical system in the 1

Universidad

Politecnica

Madrid

and

IBM Scientific

Centre

324

J.

Jimenez

neighborhood of such a manifold, but any identification describes a sizeable fraction of the time evolution of

of a low dimensional structure a flow, opens the possibility that

analysis

capture

in

some

its

of the

neighborhood

might

quantitative

features

give

of the

results

that

complete

the

qualitative

and

which a local

perhaps

even

flow.

REFERENCES BENZI,

R._

PATARNELLO,

two-dimensional L)EMAY,

S.

decaying

-Y-. _5 IOOSS,

Taylor 193-216. HERNAN,

avec

M.

turbulent JIMENEZ,

les

A.

G.

deux

_

mixing

_

1984

en

JIMENEZ, layer. the

J.

J.

1980

On

J.

1987

Bifurcations

Europhysics

Calcul

cylindres

J. Fluid

des

1987 3,

solutions

rotation.

1982

Mech.

visual

P. Left..

SANTANGELO,

turbulence.

growth

bifurcuaees

J.

Maecanique

Computer

analysis

119,

On

the

statistical

properties

of

811-818. pour theor,

le probleme appl..

de Couette-

Num.

of a high

speed

film

layer,.

J. Fluid

Special,

of the

plane

323-345.

of a turbulent

mixing

Mech.

96,

447-

460. JIMENEZ_ 30, JIMENEZ,

J.,

sublayer LANDAU,

and

bursting

in

two-

dimensional

Couette

flow.

Phys.

Fluids.

3644-3646. MOIN,

P.,

of a turbulent L.

D.

_

R.

D.

MOSER

channel.

LIFSHITZ,

These E.

M.

_

KEEFE,

L.

1987

Ejection

mechanisms

in the

Proceedings. 1959

Fluid

Mechanics.

Pergamon

Press,

p. 123.

N88-23120 Center

for

Turbulence

Proceedings

of the

325

Research Summer

Program

Coherent some

1987

structures background By

When

Fazle

Hussain

asked

us to discuss

boundary layers in modeling, my first don't want to give a talk." My second most disappointing time." After losing the

initial

question.

Since

three

from

smaller

Kline

how we can

thought thought

1

use

the

knowledge

However,

thinking

First mutual

are other

about

the question

definitions,

parts

in

perturbations;

which has interested you I still don't know

has

led

me

pain years

research describing

(b)

of refusing in as much

for an

let me call

time:

(a)

creation

this

creation

of

of smaller

coherent/p

the

would die out as the cases like the boundary

many years that anti cascade-like

If we define

It ought to have larger structures

scale

motions

been evident in turbulent

to be a qualitative

appear

shift

to be "growth-like"

example

some

a flow

localized

part

It is important that the

separation of the not

it is particularly computer

they

they

will

Rij fully

and

tend be

the

to

values

but that

to study

the

growth Such

hypotheses and being

than other instabilities

very

large and

significant

"jitter"

instabilities

from

measure probe

averages

realization

with

fixed

separations.

University

larger

intervening of the anti

sense,

advected

transfer

then

appear

order whole

in

that

frame,

processes

or higher alter the

energetic motions

not happen to downstream

a statistical

(b)

energy

and

other

of instabilities

to be

instabilities. flow field,

processes

coherent/p

grow

to

tell

as

within

and

For

will and

structures

tend

respect to ensemble data of Kim, Kline profiles

about

of

is the even the

U(y,

both has

only

and

also or

thus

in are

very rapidly from initial conditions. to

this

will

but

analytically

exponentially,

to realization.

us

different,

They change differences in

probes

of the coherent/p

structures

are

experimentally,

Moreover,

reconfirm,

Stanford

the

in the existence

in the

type

flow structures. they amplify

sources

the

in

pattern

affect

very well known mean profile U(y). Not only structures, but the implications are perhaps using

from

know that does of energy exists

typically

movie

with

something deny the

and

types

with As the

of Abernathy

least now

instabilities,

situation profile.

from the coherent/p

scales

(Kolmogorov scale). Part (a) processes. Thus cascade theories so dominated ideas

secondary do not

only

two

type

of reasons.

the ensemble

and are instabilities

rather

structures structures)

field. recognize

identify at

does,

flow

instabilities

in individual realizations. The to that of the mean velocity

concerned

1

flow,

to have

hard

appreciated,

deviate for the

to

for a number

correlation

be seen analogous

flow

difficult

less apt to satis_aylor's point to point in the Thus

often

only

to

an article to a leading journal made us delete the remark on

and we a source

in the

local

These (larger

of many

sequences of events, of the mean motion.

"breakup" (or breakdown) type local instabilities The modifier "local" is used to indicate that the for

seem

all along that something, some flows since otherwise all turbulent

flow goes to downstream infinity, layer and the mixing layer where

an instability

(a) processes

that

motions) in the flow stream, into turbulent

structures

when we submitted process, the reviewers

infinity. In those situations, there are repeatable create quasi-coherent structures from the energy type

in

hence I is the

me for a long the answer to

ideas

structures.

coherent/p

publication. So we seem to have learned at as few people in this summer institute would

cascade-like processes. process, has to create

to some

(identifiable mean flow

motions; (c) decay of the smaller scaled motions owing to viscosity is an "anti cascade-like" process; parts (b) and (c) are "cascade-like" theories can describe parts (b) and (c) but not part (a). Cascade-like in turbulence around 1970

structures

to that, and to the question

let me ask, "What are coherent structures in turbulent flows?" in understanding of what I am talking about. My personal answer

possible

identifiable

of coherent

was, "I don't know the answer was, "The lack of an answer

is: "Coherent structures are a sequence of events significant amounts of mechanical energies of the

there

have

J.

aspect of the work on coherent structures, a bit of sleep for three nights, I have to tell

be worth putting forward. order that we have some to the question which convert fluctuations."

S.

and modeling: comments

these

have not

very

reasons

little

if an

always rarely,

been if ever,

averaging is precisely and Reynolds showed t) in the

boundary

all

situation precisely analogous more important since we are details

of the

dynamics

and

326

S.

J.

Kline

thus the ensemble averages if taken too literally may mislead us. Yann Guezennec's remarks about ways to formulate the ensemble averages by folding structures with "strong left-handed" elements onto those with "strong right-handed" elements are an excellent example of the dangers and what needs to be done about them. It is also in the nature of growth-like instabilities in hydrodynamics that they respond to a range of perturbations of input parameters with a range of parametric variation in the outputs. Indeed there is now clear evidence, when one begins to put it together, as S. K. Robinson of NASA Ames and I are currently doing, to show quite clearly that there are several (more than four) quite different kinds of perturbations that can set off the growth instabilities involved in the turbulence production in a boundary layer and that at least three kinds of local instabilities occ,r. It is n,t yet clear what fraction each of the three contributes to the totality of production of turbulent kinetic energy in the wall layers. All this suggests that the coherent/p structures are not just an elephant, but are more like a zoo of various animals, each of which occurs with some variation in size and shape. It is easy to deduce examples that demonstrate this fact, but Professor Hussaln's strictures on length of this material preclude those details here. In sum the physics of turbulence production in the boundary layer is of truly great complexity, unlike most physical situations in the world which are simple once they are well mapped and understood. I summarize this "dismal view" of the problem not only because it strongly affects the way we need to conceptualize the physics, but more particularly because for this discussion it implies several things of importance about the utility of structure information in mathematical modeling. In discussing the use of structure information in modeling, it will be useful to distinguish three "levels of help" that derive from structure information: 1. Qualitative knowledge that helps guide designs including ideas for flow control: e.g., Riblets, LEBUS, wall curvature, mixing control in jets and cyclones, 2. Suggestions

about

needed

precautions

3. Direct use of the features, the sequence source for mathematical models.

in probe

measurements,

of events

that

create

in LES/FTS, turbulent

and

kinetic

. .... in modeling. energy

as the

The knowledge of the coherent/p structures has already influenced all the examples of design and control functions listed under 1, and many others not listed, although just how much is hard to say accurately. There is little doubt that the information will continue to serve this function, probably increasingly so as we gain more detailed knowledge. The knowledge of coherent/p structure at level 2 can, and sometimes has already been critical even though the number of detailed steps involved with that knowledge is small. Some examples include the difficulties with data rate and probe size in lab experiments and grid resolution in FTS that we now know occur as Reynolds number increases owing to the way the coherent/p structures scale. The most disappointing feature of the several decades of work on coherent/p structures in the boundary layer is the near total absence of use of the information for direct building of mathematical models. So far, outside of some small guidance in the wall model of kuhn and Chapman and important use of the scale information and ensemble average structure model by Perry, Henbest and Chong, there has been very little successful use of the coherent/p information in modeling. Noteworthy attempts have been made by many, including for examl?le Landahl and co-workers, by Beljaars and by Lilley, but thus far these have not led to computations of practical engineering significance. The situation remains that we do most practical modeling via some form of the RANS or equation with closure supplied by models or with even simpler equations such as There are then two critical questions. Will this situation continue, or will we be able to build a mathematical model directly on increased information about the coherent/p structures? Second, if we do not build such models, will the coherent/p information be important in the simpler models? With regard to the first question, I cannot provide an answer, as I noted at the outset; one needs a crystal ball, and mine is cloudy. Only time will tell. One can say this much, the complexities of the problem, some of which are outlined above, mitigate rather strongly against the creation of a complete usable mathematical model built directly on coherent/p information. As we learn more about details in given cases and about a wider range of cases, that becomes clearer and clearer. However, a little careful thought suggests that even if we do not use the coherent/p information to build a mathematical model directly, that information will be critical in the long run. I have

Coherent not

seen

this

enough

argument

about

the

made,

problem

and

to

lot

so long

for many

simple e_tent fact

I am

see

We need to remember that continuity (and when necessary flows

structures

the

that

the

as the

fluid

If we utilize

complete

is Newtonian-at

least

simpler

Navier-

build

a "unified

to the output,

layer

level IF

of two and

external

line

suitable

shear

, and

point

In ities

of the

of the the

which

the

I want

described,

details

of the

could

of our one

be

in

bring

brought

groups such at USC and

the

on

the

to

that

they

this

by

now

bases

questions to be

compared

of

so

more

with

knowledge

at

far

cover

more data

H. active bases

Kasagi groups, of Spalart

at

and

often

factor by

the

lead

me

may be equations

and

used

only

underestimated.

make the the genesis

point. Prandtl's of modern fluid

information

direction.

theory,

that

the

Mathematicians

including

structures

and

we have in

the

the

Theory,

conclusions.

give

their

had

uses the

so much

research

example,

reprinted

Given

in

in

the

complex-

difficulty

community

not

be

the

opposite

answer.

drastically

alter

Ames

a handful do

of

provide

cases

feasibility of magnitude

laboratory

work.

structure

likely

in

about

and

M.

More

available

matters I would

the

are

do

situation.

restricted

exemplary

specifically, in the

various

State, transfer

R.

Khabahkpasheva

questions

of Moin/Kim,

and

advance

to cases,

a decade ago and equally increase in speed and

E. Falco at Michigan and add in the heat clear

to

in central,

beyond orders

Tokyo, some

key

Hussain,

Approximation

would

NASA

layer

R. Smith at Lehigh, R. and others at Gottingen

the

grounds; it cannot. At least all and more important, the Navierbut rather use mathematics to

why

they

boundary

up

concerned government agency asked me if to get through the problem I said, "No!" I ideas or research tools, and the added man-

available

entirely several

of one

available

suggested

in content,

empirical

drawing

I would

conventional on

to as

the

and

money,

only

another

small

agreement

nevertheless

do

predict are not

axisymmetric)

Professor

streamwise

it is clear

today,

that lay done with

at Stanford,

a few of the

the computer

bear

to

is consequently

of coherent/p

reaching

all the

and

clearly

by

underlying

before

above in

predicts

use of coherent/p information be dropped in forming model

mechanics accepted

relevant

remark

Numbers,

as those of C. H. Eckelmann only

precisely to the follows from the

of more than 10data and computer model equations are made explicit

of flow

in the

a

those

in the recent dissertation been able to show that planar

show

use

length

knowledge

question

bases

data

existing

of R. J. Moffat

to mention questions

the

manpower

to that

data

Reynolds

investigating they allow

together

groups

that low

decrease

flows This

in the sense

which

relatively

Similitude

more and

the

for

is quite then

whenever we do careful testing conferences but by no means

organized

is often

The

PART,

because

a means important

only

far-field,

"zone"

session

to its

IN

so precisely

true

this

Sabin-Abramovitz

in fluid generally

dynamics

If asked

is

and

that

important can not

monograph

future

faster.

it

far,

of averaging,

quite

importance equations,

world.

to make

physics

unfortunately

not

can be justified on mathematical are circular in some subtle way, are not mathematical in content,

physical

significantly While

thus

form

a generalized

most terms

even the kinematic details. Some years ago a strongly the time was right for a push on turbulence in order did not think so because there were not enough good power,

model

in the

explicitly

in my own

about

one

hard

results

its importance

and

argued this I have seen themselves

from

information

compared

layer,

think

parameters,

analysis;

is small

thinking

finding

This

in a long

critically

are discussed in detail 1986 by Springer-Verlag.

boundary

some

discrepancy in the k-e

These

The what

depends

a model

we have

via

and

governing

discussions

thickness

have sometimes such arguments Stokes equations provide

tile

an example of central of the boundary layer

layer

me

Navier-Stokes equations plus excellent model of turbulent

an

equations

near-

with no constants

parameter.

problems.

at one

mechanics,

making

the totality of turbulent or simplifying process.

k-e model

layers;

of variation in the physics about a "zoo of animals".

for particular give

data

typically

are a good

the

mixing

an important conclusion. critical decisions about

for a moment Let me derivation

using

jets,

stretching

of thought

to what seems in making the

for

current example appears In this work Tzuoo has

non-dimensional

a vortex

is just the kind metaphor above This

model"

(wakes,

of profiles of Reynolds stress AND ONLY IF the internal

parameter

for all the

equations

Hussain

clearly.

is borne out again and again and again in the 1968 and 1980-81 AFOSR-Stanford

zonal

cases

functions

more

327

honest-to-God equation) are

equations

limited to those examples. An enlightening S. Tzuoo with J. H. Ferziger and the writer. free-shear

Professor

for modeling in the averaging

Stokes

modeling

of Newtonian fluids, but also that they describe no mode; they do not suggesting behavior that cannot occur. Hence, the losses of information

trivial. This non-triviality of models as, for example,

can

to

a little

the full, unaveraged, the viscous energy

cases.

the motion information

grateful

results

equations must be insufficient that we have lost information

describe ezcess

and

emerge. as a number

when

we

research

Blackwelder results in the

in Novosibirsk, When

we take

of workers

these in this

328

S. J. Kline

conference, including S. K. Robinson (whose dissertation will focus on these question), have begun to do, answers very quickly begin to emerge. Some of these results answer questions that have been troubling me for a decade or two, and more important, the reasons why we had not been able to resolve them before are also clear. All this is very promising and augurs very well for rapid progress over the next few years on the question of the kinematics and dynamics of coherent/p structures in turbulent boundary layers. CONCLUSIONS 1 tt_A,.rs* _d;.::g ef *.he phys;_c_ c-_.--._ _._-.:'r_.. H:.'-_ry and sometimes surprising ways.

_e!!_ ".:'_"'_ c.!:',,.c._'_c.!:_-_.y_ "h_1_"

i,, i,_},v, L_.L,

2. "Help" has been and may remain more at levels 1 and 2 than at level 3. But even at level 3, in direct mathematical modeling, the use of coherent/p information may be critical in deciding what terms can not be dropped in simplifying model equations for particular flow zones. 3. These coherent/p

potential gains structures.

more

than

justify

the current

level of research

efforts

on understanding

iT ;.88: 23 12 z, : Center

for

Turbulence

Proceedings

of the

329

Research

Summer

Program

1987

Coherent By

In order ative

to develop

value

observer. value can

over

more

a range

Structures

CHAItLES

quantitative

(3.

measures

of experiments,

SPEZIALE

of coherent

it is essential

that

=,_+,_ (where

_ is the

serves

this

mean,

purpose.

a dominant

_bc is the For

temporal

coherent

a statistically

frequency

part,

f we can

=

lim T-*co

theories

with

_b_ is the Hussain

1

T

--I

random

which

(see

r

measures

would

be

have

compar-

independent

a fundamental

of the predictive

(1)

turbulence

take

that

such

+_,

and

steady

structures

that

It is only through such a general framework be developed. The triple decomposition

1

part

of any

possesses

turbulent

coherent

field

structures

_b)

with

1983)

(2)

JO

_(X,

t)dt

N _be = where


denotes

subject

to

random

a suitable the parts

be noted that parts of any should

of the

event

some will

be the

double decompositions turbulent field ¢ that,

therefore

only

be used

of _bI (i.e., E.

when

Op

NASA

Langley

Research

same

the

an such

mean

flow

ensemble triple

for all observers 1983, depend flow

Otherwise,

for the mean and coherent form (see Hussain 1983):

Dt -

With

(see Hussain in general,

the coherent notion (see Speziale 1986). that are overly biased by the observer.

The equations of motion can be written in the

ICASE,

average

of

turbulence

D_i

1

conditional

occurrence

(5)

fields,

based

(see

Speziale

1986) give rise to on the observer.

vanishes one

average

runs on the

over

flow

decompositions,

or is negligibly the

risk

triple

the 1986). coherent Double small

of extracting decomposition

0

0_, +,.v'a,Center

_-(,,._uc,

+_)

(6)

330

C. G. Speziale

Duci

Opc

Dt

-

Of_i

Oz-'--i,+ uV2

Ouc

uc" - u,_ Ozj

ucj

Ozj

(7) DfLi

Op

Dt

Omi

0 + uV2_i

where the

v is the

velocity

kinematic

field

viscosity

which

of the

is subject

to

fluid,

the

-

p is the

continuity

-Ozj

< urlu_j

modified

equation

>

pressure,

which

and

yields

the

u = _ + Uc +

u_

v._=o

(8)

W'_c

=0

(9)

v. a_ = o In order

to

achieve

closure

of the

equations

need

to be modeled.

popular

The

two-equation

time-averaged

models

transport

hypothesis

(see

Hussain

can

be

aries, to

considered the

where

large-eddy

model

simulations, the level simulation

of

and

problems

(e.g.,

for

some

dimensional. new

In my

information

along

there the

these

eddy This

advantage

turbulent

opinion,

is

nature

url

u,. i can

Reynolds

be modeled

(see

viscosity

(_ Ouc, O_Cj

--liT

tried).

computation since a coarse

concerning

investigations

an

_>:

be

the

stress

terms

(11)

models

eddy

appropriate

can

has

Furthermore, rect numerical

stress closure

Hence,

UriUr'i

UT is an

Smagorinsky

(10),

Launder,

using Reece

the and

currently Rodi

1975

Reynolds stress < uri u_j > primarily serves as an energy it is plausible that it could be modeled using a gradient

1983).


, urlur_

Reynolds

or second-order

and Lumley 1978). The phase-averaged drain on the coherent motion and thus

(lO)

of motion

< uriur_

is

constraints

models

Ouc_

"l'-

viscosity approach,

in that

the

mixing

layers) chance

of coherent

(12)

)

(sufficiently

far

a certain

coherent

the

structures

bears

motion

such and

uc

from

solid

bound-

resemblance

is calculated

directly

less than that needed for a difine-scale turbulence is modeled)

coherent

that

form

which

required is substantially mesh can be used (the a good

of the

motion an

approach

it is well

is approximately could worth

yield

pursuing

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